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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