Let $f: \Bbb{R}\to \Bbb{R}$ and $g: \Bbb{R}\to \Bbb{R}$ be continuous functions, where $g(x) = 0$ for all $x \notin [a,b]$ for some interval $[a,b]$.
a) Show that the convolution $$(g*f)(x):=\int_{-\infty}^{\infty} f(t)g(x-t) \,dt$$ is well defined.
I understand how to use change of variables to simplify the problem to a non-infinite integral, but I am not sure how to show that it is well-defined. I am struggling to understand how the definition of well-defined and how to show it.
b) Suppose $$\int_{-\infty}^{\infty} \lvert f(t) \rvert \,dt < \infty$$ Prove that $\lim_{x\to-\infty} (g*f)(x)=0$ and $\lim_{x\to\infty} (g*f)(x)=0$.
I understand that there is some relation to a sequence/series which should make this straightforward, however, I cannot figure out how to get there.
Any help would be greatly appreciated!