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This question deals with why the derivative of $f$ is not defined at discontinuities in $f$. I found the answers satisfactory.

My question deals with why the derivative is not defined at discontinuities in itself. Take the following piecewise function:

$$f(x) = \begin{cases} g_1(x) & x < 1 \\ g_2(x) & x \ge 1 \end{cases} = \begin{cases} x^2 & x < 1 \\ x & x \ge 1 \end{cases}$$

This function is continuous, but its derivative is not. The function $g_2(x)$ has a derivative at $x = 1$, yet continuously gluing it with $g_1(x)$ apparently removes this tangent. Why is $f'(1)$ undefined instead of $f'(1) = 1$?

user110391
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  • Simply because the limit $\lim_{x \to 1} \frac{f(x)-f(1)}{x-1}$ does not exist. – Martin R Sep 20 '23 at 12:02
  • For the derivative to exist, one would have to get the same value for the derivative from the right (when $h$ is positive) and the derivative from the left (when $h$ is negative). Here you get two different values. – Mikhail Katz Sep 20 '23 at 12:17
  • @MartinR I think I get it now. The derivative at a point is defined as a limit as $x$ goes to that point, meaning the properties of having a discontinuity at a value and being undefined at a value are the same for a derivative, because the $\lim_{x \rightarrow a}(\lim_{x \rightarrow a} f(x) ) = \lim_{x \rightarrow a} f(x)$. Therefore, they are equivalent properties. – user110391 Sep 20 '23 at 13:41

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The derivative of f(x), at $x= x_0$ is defined as $\lim_{h\to 0}\frac{f(x_0+ h)- f(x_0)}{h}$. Obviously, the denominator of that fraction is going to 0. In order for the limit to exist, the numerator, $f(x_0+ h)- f(x_0)$ must also go to 0 which means that f must be continuous at $x= x_0$.

(f being continuous is a "necessary" but NOT "sufficient" condition for f having a derivative. f(x)= |x| is continuous at x= 0 but does not have a derivative there.)

George Ivey
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    It IS an anwer to the question asked! That was, why functions cannot be differentiable where they are not continuous. I added the remark about continuity not being sufficient because I happened to think about it. I did not prove it because it was not asked! – George Ivey Sep 21 '23 at 00:45
  • Apologies, I misread your answer. If you make an edit to your answer, I will be able to replace my downvote with an upvote. Currently I am not allowed by the site. – user110391 Sep 26 '23 at 18:07