Differentiability implies continuity

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We see that if a function is differentiable at a point, then it must be continuous at that point.

There are connections between continuity and differentiability.

Differentiability Implies Continuity If $f$ is a differentiable function at $x = a$ , then $f$ is continuous at $x=a$ .

To explain why this is true, we are going to use the following definition of the derivative $f'(a) = \lim _{x\to a} \frac {f(x)-f(a)}{x-a}.$

Assuming that $f'(a)$ exists, we want to show that $f(x)$ is continuous at $x=a$ , hence we must show that $\lim _{x\to a} f(x) = f(a).$ Starting with $\lim _{x\to a} \left (f(x) - f(a)\right )$ we multiply and divide by $(x-a)$ to get

$\begin{align*} &= \lim_{x\to a} \left((x-a)\frac{f(x) - f(a)}{x-a}\right) \\ &= \left(\lim_{x\to a} (x-a) \right) \left(\lim_{x\to a}\frac{f(x) - f(a)}{x-a}\right) &\text{Limit Law.} \\ &= \answer[given]{0}\cdot f'(a) = \answer[given]{0}. \end{align*}$ Since $\lim _{x\to a}\left (f(x) - f(a)\right ) = 0$ we see that $\lim _{x\to a} f(x) = f(a)$ , and so $f$ is continuous at $x=a$ .

This theorem is often written as its contrapositive:

If $f(x)$ is not continuous at $x=a$ , then $f(x)$ is not differentiable at $x=a$ .

Thus from the theorem above, we see that all differentiable functions on $\RR$ are continuous on $\RR$ . Nevertheless there are continuous functions on $\RR$ that are not differentiable on $\RR$ .

Which of the following functions are continuous but not differentiable on $\RR$ ? (Select all correct answers)

$x^2$ $\lfloor x \rfloor$ $|x|$ $\frac {\sin (x)}{x}$

From our informal definition of derivative of a function, we can see that the piece-wise function

$f(x) = \begin {cases} 1 &\text {if $x\leq 3$,}\\ 0.5(x-3)+1 &\text {if $x>3$} \end {cases}$ is not differentible at $x=3$ :

Notice that however much we ’zoom in’ on the function at $x=3$ , there is always a kink.

Using the ’zoomed in’ graph of the piecewise function $f(x)$ above, what is $f'(3)$ ?

$f'(3)=0$ because to the left of the ’kink,’ the line is
horizontal, which has slope $0$ . $f'(3)=0$ because to the right of the ’kink,’ the line has slope $0.5$ . $f'(3)=0.25$ because $0.25$ is the average of the slopes of the lines to the left and right of the kink. $f'(3)$ does not exist because there is no one
best tangent line approximation of the function at $x=3$ .

Using that limit vocabulary, explain why the piecewise function $f(x)$ is not differentiable at $x=3$ . It may be helpful to note that the slope of $f(x)$ when $x<3$ is $0$ and the slope of $f(x)$ when $x>3$ is $0.5$ .

Consider $f(x) = \begin {cases} x^2 &\text {if $x<3$,}\\ mx+b &\text {if $x\ge 3$.} \end {cases}$ What values of $m$ and $b$ make $f$ differentiable at $x=3$ ?

To start, we know that we must make $f$ both continuous and differentiable. Hence, we must ensure that the value of both pieces of $f$ agree at $x=3$ . Write with me $\begin{align*} \eval{x^2}_{x=3} &= \eval{mx+b}_{x=3}\\ 9 &= m\cdot 3 + b. \end{align*}$ Now we must ensure that the derivatives of each piece of $f$ agree at $x=3$ . Write with me $\begin{align*} \ddx x^2 &= \lim_{h\to 0}\frac{(x+h)^2-x^2}{h}\\ &= \lim_{h\to 0}\frac{x^2 + 2xh + h^2 - x^2}{h}\\ &= \lim_{h\to 0}\frac{2xh + h^2}{h}\\ &= \lim_{h\to 0}\left(2x + h\right)\\ &=\answer[given]{2x}. \end{align*}$ Moreover, $\ddx (mx+ b) = \answer [given]{m}$ by the definition of a tangent line. Hence we must have $\begin{align*} \eval{\ddx x^2}_{x=3} &= \eval{\ddx(mx+b)}_{x=3}\\ \eval{2x}_{x=3} &= \eval{m}_{x=3}\\ 6 &= m. \end{align*}$ Ah! So now $\begin{align*} 9 &= m\cdot 3 + b\\ 9 &= 6\cdot 3 + b\\ 9 &= 18 + b, \end{align*}$ so $b=-9$ . Thus setting $m=6$ and $b=-9$ will give us a continuous and differentiable piecewise function.