I'm pretty new to convolutions and I had to evaluate the convolution $\mathbb{1}_{[0,1]} * \mathbb{1}_{[0,1]}$
I found the following solution: $$\mathbb{1}_{[0,1]} * \mathbb{1}_{[0,1]} (x) = \begin{cases} 0 \qquad x \le 0 \\ x \qquad 0<x<1 \\ 1 \qquad x=1 \\ 2-x \quad 1<x<2 \\ 0 \qquad x\ge 2 \end{cases}$$
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I know the following theorem holds:
For every $f,g \in C_c(\mathbb{R}^n)$ we have $\ \text{supp}(f*g) \subseteq \text{supp}(f) + \text{supp}(g)$
But as you can see the two functions above are not continous but anyway the theorem seems to hold. My question is:
How can I generalize that theorem? In other words, what are the minimalist hypothesis on $f$ and $g$ such that $\ \text{supp}(f*g) \subseteq \text{supp}(f) + \text{supp}(g)$ ?
Continuity is not required at all really, but you need to be careful about the word "support", because evaluating functions $f \in L^1$ is not quite kosher (technically, functions in $L^1$ are defined only up to a.e. equality.) However, if $f, g \in L^1$ are supported on sets $X, Y$, meaning $f \equiv 0 $ in $X^c$ and $g \equiv 0$ in $Y^c$, then $f * g$ is supported on $X+Y$. This is immediate from the definition of convolution. See also https://en.wikipedia.org/wiki/Support_(mathematics)#Essential_support.