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Wize AP Calculus (BC) Textbook > Limits & Continuity

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
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}