# Is it true that differentiable functions can have essential discontinuity

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.

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.

• 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. Apr 2 '16 at 12:40
• 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. Apr 2 '16 at 17:48

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.

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 $$0$$.