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Consider the function $$f(x)=\left\{\begin{array}{cc} x^2-4 & \text{ if }x\leq 2\\4x+3&\text{ if } x\gt 2\end{array}\right.$$

This function is differentiable at $x=2$ since $\lim_{h\to 0^{\pm}}\frac{f(2+h)-f(2)}{h}=4$ (EDIT: this actually isn't true but it is true that $\lim_{x\to 2^-}f^\prime (x)=4=\lim_{x\to 2^+}f^\prime(x)$); however, it's not continuous at $x=2$.

How is that possible, doesn't differentiability at $x=a$ imply continuity at $x=a$?

This question came up when I tried to answer the question of finding $a$ and $b$ such that the function $$f(x)=\left\{\begin{array}{cc} ax^2-b & \text{ if }x\leq 2\\bx+3&\text{ if } x\gt 2\end{array}\right.$$

The solution is achieved by finding conditions on $a$ and $b$ such that it's continuous, and also such that the left/right derivates exist. The left/right derivative question gives $4a=b$. With the condition of continuity you get the additional condition that $4a-b=2b+3$, giving a unique solution. But doesn't differentiability imply continuity? What's wrong with just solving $4a=b$ like in the first example above?

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  • $\begingroup$ It is more so that differentiability requires continuity rather than implies it. $\endgroup$ Oct 26 '20 at 20:15
  • $\begingroup$ If I can't assume that $\lim_{x\to 2}f^\prime(x)=\lim_{x\to 2}\frac{f(x)-f(2)}{x-2}$, then how do I solve my second question? The solution uses that $\lim_{x\to 2^-}f(x)=\lim_{x\to 2^+}f(x)$ and that $\lim_{x\to2^-}f^\prime(x)=\lim_{x\to2^+}f^\prime(x)$...how does that work? Why can we get away without using the difference quotient? $\endgroup$
    – user162520
    Oct 26 '20 at 20:25
  • $\begingroup$ It is simply the premise of the problem. Since you don't know whether or not the function is continuous at $x=2$, before you find differentiability at $x=2$, you must first account for its continuity. Compare this to a problem that begins with something like Let $F:\Bbb R\to\Bbb R$ be a differentiable function such that... (in which case you can assume $F$ to be continuous). $\endgroup$ Oct 26 '20 at 20:31
  • $\begingroup$ Your limit computation at $2^+$ is wrong. It is not $4$. $\endgroup$
    – user65203
    Oct 26 '20 at 20:33
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    $\begingroup$ @user162520 I am editing my answer. But I would like you to calculate the RHD. You will see that the RHD exists iff $4a=3b+3$. $\endgroup$ Oct 26 '20 at 22:21
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Note that $f(2)=0$ and that therefore$$\lim_{x\to2^+}\frac{f(x)-f(2)}{x-2}=\lim_{x\to2^+}\frac{4x+3}{x-2}=\infty.$$So, $f$ is not differentiable at $2$.

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The function is not differentiable at $2$ because $RHD$ is $$\lim_{h\to0^+}\frac{f(2+h)-f(2)}h=\lim_{h\to0^+}\frac{[4(2+h)+3]-0}h\to\infty$$Note that it is only the LHD which is $4$.


It is given that the function is differentiable at $2$. $$\begin{align*}LHD&=\lim_{h\to0^+}\frac{f(2)-f(2-h)}h=\lim_{h\to0}\frac{4a-b-[a(2-h)^2-b]}h=4a\\RHD&=\lim_{h\to0^+}\frac{f(2+h)-f(2)}h=\lim_{h\to0^+}\frac{2b+hb+3-[4a-b]}h\end{align*}$$

Note that for the $RHD$ to exist, the limit must have the $0/0$ form since the denominator tends to $0$. The numerator will tend to $0$ iff $2b+3-[4a-b]=3b+3-4a=0$, giving you the $RHD=b$.

Equating the $RHD$ and $LHD$ gives $4a=b$ and existence of $RHD$ requires $4a=3b+3$ which is also the condition which we obtained from analyzing the continuity of $f$. Thus there is no need to consider continuity separately.

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  • $\begingroup$ Why can't I just say that $\lim_{x\to 2^-}f^\prime(x)=4$ and also $\lim_{x\to 2^+}f^\prime(x)=4$ $\endgroup$
    – user162520
    Oct 26 '20 at 20:08
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    $\begingroup$ That is true, but note that $RHD=\lim_{x\to2^+}\frac{f(x)-f(2)}{x-2}\ne\lim_{x\to2^+}f'(x)$ in general. In fact $f'(x)$ does not exist at $x=2$ even though the limits as you observed exist. $\endgroup$ Oct 26 '20 at 20:09
  • $\begingroup$ that blows my mind! Why doesn't $\lim_{x\to2^+}f^\prime(x)=\lim_{x\to2^+}\frac{f(x)-f(2)}{x-2}$ $\endgroup$
    – user162520
    Oct 26 '20 at 20:11
  • $\begingroup$ The LHS in your expression gives the limit of the derivative from the right of $2$ and assuming differentiability, the RHS gives the value of the derivative at $2$. For discontinuous $f'$, they may not agree. The same happens with $y=|x|$. The function is not differentiable at $0$ but the RHL of $f'(x)$ is $1$. $\endgroup$ Oct 26 '20 at 20:14
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    $\begingroup$ The LHL of $f^\prime(x)$ is $4a$ because $b$ is a constant. $\endgroup$
    – user162520
    Oct 27 '20 at 12:52

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