মিথ্যা কনভার্স: অবিচ্ছিন্নতা মানেই অন্তরীকরণযোগ্যতা নয়

First take a moment to consider why a continuous function might not have a derivative.

What kinds of behavior would prevent a function from having a derivative?

Here's one example of a function that is continuous everywhere that doesn't have a derivative at 0.

Intuitively, $|x|=x$ for $x \ge 0$, so for $x>0$ the derivative is one. Similarly, since $|x| = -x$ for $x \le 0$, the derivative is negative one for $x<0$. So what happens at $0$?

Essentially we get two conflicting values for the derivative at $0$ depending on the direction from which we approach zero in the limiting process for the derivative. Since neither one is "better" than the other, there is no way to get a derivative at zero, and it simply doesn't exist.

There are even functions that are continuous everywhere, but differentiable nowhere.

Click here for one example