Convolution of a piecewise function with itself

Question

Convolution of a function with itself I was going through this question. In the answer, the limits of the integral was transformed from (-infinity)-(+infinity) to 0 to x. Can anyone explain how this happened?

Answer 1

The function is non-zero only in $[0,1]$. So, for the convolution to be non-zero you need $$ 0 < x - y < 1 \implies x-1<y<x \ $$ $$ \ \text{and} \ 0<y<1.$$ If $0<x<1$ then clearly $$0<y<x$$ has to hold by the two inequalities above.