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Wize University Calculus 1 Textbook > Limits

Continuity

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Continuity

In calculus, we think of a function being continuous if we can draw its graph without picking up the pencil as we draw. A function will be continuous if it has no breaks or gaps.

Continuity at a Point

A function ff is said to be continuous at a point aa of its domain if all of the following conditions are satisfied:

a) f(x) is defined at x=a \text{a) }f(x) \text{ is defined at x=a }

b) lim⁡x→a f(x) exists\text{b)} \ \displaystyle\lim_{x\rightarrow a}\ f(x) \text{ exists}

c) lim⁡x→af(x)=f(a)\text{c)}\ \displaystyle\lim_{x\rightarrow a}f(x)=f(a)


Continuity on an Interval

A function ff is said to be continuous on an interval (a,b)(a, b) if it is continuous for every cc in(a,b).(a, b).

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Types of Discontinuities

1. Asymptote







2. Jump





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3. Hole






4. Oscillating


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Example: Continuity


Consider f(x)=sin⁡2xx\displaystyle f(x)=\frac{\sin{2x}}{x} . Does lim⁡x→0sin⁡2xx\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}}{x} exist? Is ff continuous at x=0 ?x=0\ ?

lim⁡x→0sin⁡(2x)xRecall: lim⁡x→0sin⁡(kx)kx=1=lim⁡x→02sin⁡(2x)2x=2lim⁡x→0sin⁡(2x)2x=2\begin{array}{l}\displaystyle\lim_{x\rightarrow 0}\frac{\sin(2x)}{x}&&\text{Recall:}\ \displaystyle\lim_{x\rightarrow0}\frac{\sin(kx)}{kx}=1\\\\=\displaystyle\lim_{x\rightarrow0}\frac{2\sin(2x)}{2x}\\\\=2\displaystyle\lim_{x\rightarrow0}\frac{\sin (2x)}{2x}\\\\=2\end{array}

The function is not continuous at x=0x = 0 because it is not defined at this point ( x=0x = 0 makes denominator to be equal to zero!)



Determine the values of aa and bb such that f(x)f(x) is continuous, where


f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}


Extra Practice
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)\left(a,b\right).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)(a,b).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Use the graph below to find
Continuity
Use the graph below to find
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Practice: Evaluating limits
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.
f(x)f\left(x\right) is discontinuous at x=0
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Find values of aa and bb such that the following function ff is continuous at x=0x=0 and x=1x=1:
f(x)={a(1−cos⁡2xsin⁡(2x)), if x<02x2−x+b, if 0≤x≤1x2+2x−3x2−1, if x>1f(x)=\begin{cases} a\left(\dfrac{\sqrt{1-\cos^2x}}{\sin(2x)}\right)&,\text{ if }x<0\\\\ 2x^2-x+b&,\text{ if }0\le x\le1\\\\ \dfrac{x^2+2x-3}{x^2-1}&,\text{ if }x>1 \end{cases}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={x2+10;x<1−x+7a;x≥1f(x)=\begin{cases} x^2+10;&x<1\\ -x+7a;&x\geq1 \end{cases}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={−2x2+7;x<−43x+a;x≥−4f(x)=\begin{cases} -2x^2+7;&x<-4\\ 3x+a;&x\geq-4 \end{cases}
Continuity and Piecewise Functions
What values of aa and bb make the following function continuous?
f(x)={ax2−4;x<−2ax+b;−2≤x≤3ax2+bx+10;x>3f(x)= \begin{cases} ax^2-4;&&x<-2\\\\ ax+b;&&-2\leq{}x\leq3\\\\ ax^2+bx+10;&&x>3 \end{cases}
Continuity
Determine whether the following function is continuous at x=0x = 0 using the definition of continuity. You must full justify your solution.
g(x)={x2+4−2x , x<00 , x≥0\displaystyle g(x) = \begin{cases} \frac{\sqrt{x^2 + 4} - 2}{x} & \text{ , } x < 0 \\ 0 & \text{ , } x \geq 0 \end{cases}
Continuity
Let f(x)f(x) be a continuous function on the open interval (a,b)(a, b). Which of the following four statements are always true.
Select all that apply.
Continuity
Where is f(x)=cos⁡(πx2)1−x2f(x) = \frac{\cos\left(\frac{\pi x}{2} \right)}{\sqrt{1 - x^2}} continuous?
Infinite Limits and Continuity
Given the function f(x)={arctan⁡x+π/2if x≤02xif 0<x≤e1ln⁡xif x>ef(x)=\begin{cases} \arctan x+\pi/2&\text{if }x \le 0\\ \frac{2}{x}&\text{if }0<x\le e\\ \frac{1}{\ln x}&\text{if }x>e \end{cases}, which of the following statements is true about f(x)f(x)?
f(x)={a2x2+1if x<1x+4if x=1−5(a+x)if x>1f\left(x\right)= \begin{cases} a^2x^2+1&if\space x<1\\ x+4 & if \space x=1\\ -5(a+x)&if~x>1 \end{cases}
a) For what value(s) of aa does lim⁡x→1f(x)\displaystyle \lim_{x\to1}f(x) exist?
b) For what value(s) of aa is f(x)f(x) continuous at x=1x=1?
Continuity
Use the graph below to find
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Practice: Continuity of Piecewise Functions
Consider the function
f(x)={ln⁡(x2)+xif  x<−1cos⁡πx2if  −1≤x<12exif  x≥12f\left(x\right)= \begin{cases} \ln(x^2)+x & \text{if }\space x<-1\\ \cos \frac{{\pi}x}{2}&\text{if }\space -1\le x<\frac{1}{2}\\ e^{x}&\text{if }\space x\ge \frac{1}{2} \end{cases}
Limits: Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1 if x≠−2, x≠24 if x=25 if x=−2f(x) = \begin{cases} \frac{x^3-8}{x^2-4}+1 & \text{ if } x\neq -2, \, x \neq 2 \\ 4 & \text{ if } x = 2 \\ 5 & \text{ if } x = -2 \end{cases}
Limits: Continuity
Find A and B so that the following function is continuous at 0 and 1:
f(x)={A(1−cos⁡2xsin⁡2x)x<02x2−x+B0≤x≤1x2+2x−3x2−1x>1f(x) = \begin{cases} A\left(\frac{\sqrt{1-\cos^2 x}}{\sin 2x}\right) & x < 0 \\ 2x^2-x+B & 0 \leq x \leq 1 \\ \frac{x^2+2x-3}{x^2-1} & x > 1 \end{cases}
Enter your answer as A,B. For instance, if you think A=0, B=1, enter 0,1
If f(x)f(x) is continuous at x=−2x=-2 and if lim⁡x→−2(2f(x)−x)2∣f(x)∣2+1=4\lim\limits_{x\rightarrow -2}\frac{(2f(x)-x)^2}{|f(x)|^2+1}=4, then
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Find all number(s) C that make(s) f(x) continuous.
f(x)={x−C1+Cx≤02x2+Cx>0f(x)=\begin{cases}\displaystyle\frac{x-C}{1+C}&x\leq0\\2x^2+C&x>0\end{cases}
Discuss where the function is continuous: f(x)=sin⁡(x+cos⁡x)f(x)=\sin(x+\cos x).
If f(x)f(x) is continuous at x=2x=2 and if lim⁡x→23f(x)+2xf(x)−2=4\lim\limits_{x\rightarrow 2}\frac{3f(x)+2}{xf(x)-2}=4, then
Continuity
Find aa such that f(x)f(x) is continuous at x=1x = 1
f(x)={4x2−3ln⁡xx≥13x3−4axx<1f(x)= \begin{cases} 4x^2 - 3\ln{x} & x \ge 1 \\ 3x^3 - 4ax & x < 1 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuity
Discuss where the following function is continuous:
h(x)=log⁡(x+1−2)h(x)=\log(\sqrt{x+1}-2)
Consider f(x)=sin⁡2xx\displaystyle f(x)=\frac{\sin{2x}}{x} . Does lim⁡x→0sin⁡2xx\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}}{x} exist? Is ff continuous at x=0x=0?
When is the following function continuous: g(x)=x2−1x2−1\displaystyle g(x)=\frac{x^2-1}{x^2-1}?
Where is the following function continuous?
f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
Find aa,bb such that f(x)f(x) is continuous everywhere.
f(x)={x2+x+ax>1x−b0≤x≤1x3−4x+5x<0f(x)=\begin{cases}x^2+x+a&x>1\\x-b&0\leq x\leq1\\x^3-4x+5&x<0\end{cases}
🦊 TRICKY! Find A > 0 and B so that the following function is continuous at x = 0 and x = 1:
f(x)={x2tan⁡2Axifx<03x2+2x+Bif0≤x≤13x2−7x−20x2−3x−4ifx>1f(x)=\begin{cases} \frac{x^2}{\tan^2Ax}&\text{if}&x<0\\ 3x^2+2x+B&\text{if}&0\leq x\leq 1\\ \frac{3x^2-7x-20}{x^2-3x-4}&\text{if}&x>1\end{cases}
Continuity
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Limits: Continuity of piecewise functions
Identify the type of discontinuity (or state the function is continuous) at x=5,10,15x=5, 10, 15. Is there another point of discontinuity in the domain other than at the stated x-values?
f(x)={x+3x−6x−6+7ln⁡(15−x)sin⁡xif−∞<x≤5if 5<x≤10if10<x<15if15≤x<∞f(x)=\begin{cases} x+3\\ \frac{x-6}{x-6}+7\\ \ln(15-x)\\ \sin x \end{cases}\begin{array}{l} if -\infty < x \leq 5\\ if\ 5 < x \leq 10\\ if 10 < x< 15\\ if 15 \leq x < \infty \end{array}
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Continuity
Find the value of aa such that the function
f(x)={x2+1if x≤−3ax−1if x>−3f\left(x\right)= \begin{cases} x^2+1&if\space x\le -3\\ ax-1 & if \space x>-3 \end{cases}
is continuous everywhere.
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right), which is defined only defined on [−5,9][-5,9]. Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\dfrac{1}{\ln\left(f\left(x\right)\right)} is continuous.
Practice: Evaluating limits
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.
f(x)f\left(x\right) is discontinuous at x=0
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Each question is worth 3 marks.
Continuity
What value of aa makes the following function continuous?
f(x)={2x+4,x≤2ax2+x−2,2<xf(x) = \left \{ \begin{array}{ll} 2^x + 4, & x \leq 2\\ ax^2 + x - 2, & 2 < x \end{array} \right .
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
When is the following function continuous?
e3x+1sin⁡(πx)\displaystyle \frac{e^{3x}+1}{\sin(\pi x)}