We apply the same procedure that we used to obtain the definite integral. We will try to create a continuous function H by connecting "pieces" of the function F, which will be on each interval shifted exactly so that the pieces fit at points (2k + 1).

An easy calculation yields

We see that the size of the jump is always the same. Here is how F looks like:

At each node (2k + 1) we will have to subtract the size of this jump again and again, so that we obtain a continuous function. This means that in the second node we will have to subtract the jump size twice (once to balance the first jump, again to balance the second jump) and so on. Applying this procedure we obtain the function

which is the desired antiderivative on the whole real line. This is proved in the same way as we did earlier with G at the point .