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
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
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 .