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I was reading wiki and found this statement. Quoting it:

A function f is said to be continuously differentiable if the derivative f'(x) exists and is itself a continuous function. Though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.

I am confused with this statement as we know :

Function is differentiable in its domain -> continuous in that domain

Not continuous -> Not differentiable

How does a function still be differentiable even if it is not continuous ?

How It can have essential discontinuity ? I did not get it from wiki example.

Please clarify :)

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The passage in question is saying that the derivative has an essential discontinuity, not the original function. That is, it is possible to have a differentiable function $f$ such that the function $f'$ has an essential discontinuity. Of course you are correct that if $f$ is differentiable, then $f$ must be continuous.

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  • $\begingroup$ could you plz explain "The derivative of a differentiable function never has a jump discontinuity " . Is this also talking about f'(x) ? why does that make sense that the derivative of a differentiable function can have essential discontinuity where it cant have jump discontinuity. $\endgroup$ – ViX28 Apr 2 '16 at 12:40
  • $\begingroup$ Here's the rough idea. Suppose that as $x$ approaches $a$ from below, $f'(x)$ is always less than $0$. Then $f'(a)$ will have to be at most $0$, since $f$ is a decreasing function as you approach $a$ from the left. This means that $f'$ can't "jump up" at $a$, and similar arguments show that $f'$ cannot have any sort of jump discontinuity. But this argument doesn't rule out essential discontinuities, and it turns out that they are possible. $\endgroup$ – Eric Wofsey Apr 2 '16 at 17:48
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Here are some points which may be helpful for you.

  1. $f(x)$ is derivative $\to$ $f(x)$ is continuous; however the converse doesn't hold, for example, $y=|x|$ at $x=0$.

  2. A function $f$ is said to be continuously differentiable: $f$ is derivative and $f'(x)$ is continuous.

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No. A necessary condition for $f(x)$ to be differentiable that it be continuous.

It is straightforward to visualize if you have a function, say, $f(x) = \frac{1}{x}$, which is discontinuous at $x = 0$, and then augment it with $f(0) = k$ (for some Real $k$ of your choosing), you still have a discontinuous function at $x=0$ even though there is now an essential singularity at there. As such, $f(x)$ cannot be differentiable at $0$.

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