Suppose that $v=v(t,x)\in C^1([0,+\infty]\times\mathbb{R})$ is a compactly supported function. Is it true that $$v(0,x):\mathbb{R}\to \mathbb{R}$$ is Riemann integrable over $\mathbb{R}$?
Certainly $v(0,x)$ is a $C^1$ function and also it is bounded (in particular, $v(0,x)$ is zero outside a bounded subset of $\mathbb{R}$). Is there any theorem that allows me to state that $v(0,x)$ is integrable over $\mathbb{R}$?
Thanks a lot in advance.
If $f:\mathbb R \to \mathbb R$ is any continuous function such that $f(x)=0$ for $|x| >N$ for some $n$ then $f$ is surely Riemann integrable on $\mathbb R$ and $\int_{-\infty}^{\infty} f(x)\, dx =\int_{-N}^{N} f(x)\, dx$