Existence of antiderivative of discontinuous functions

It is well-known that every continuous functions admits antiderivative. In this note, we show how to prove existence of antiderivative of some discontinuous functions.

A typical example if the following function

By taking to different sequences and we immediately see that is discontinuous at . However, we will show that admits as its antiderivative.

To this end, we first consider the following function

First we show that is differentiable. Clearly whenever , we obtain

At , we easily check that

thanks to . Thus, we have just shown that is differentiable with derivative

Then we decompose into two parts as follows with

Observe that is continuous, hence there exists its antiderivative , i.e. for any . Hence, in view of the decomposition above, there holds

In other words, we have just proved that is the antiderivative of the function .

Remark: In this problem, the antiderivative of cannot be written in terms of elementary functions, so as the function . The only point in this problem is that we convert the discontinuous function (at $late x=0$) to a continuous function so that its antiderivative does naturally exist.

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