Smoothness of parameter-dependent integral implies smoothness of integrand

Question

Let $f:\mathbb{R}^d\times\mathbb{R}^n\to\mathbb{R}$ be continuous. Suppose that $$ x\mapsto\int_{\mathbb{R}^n}\varphi(y)f(x,y)\,dy $$ is continuously differentiable for each $\varphi\in\mathcal{C}_c^\infty(\mathbb{R}^n)$. Is it true that then also $x\mapsto f(x,y)$ is continuously differentiable for each $y\in\mathbb{R}^n$?

Answer 1

Expanding on Thomas’s idea, take $n=d=1$ and $f(x,y)=(y-x)^+$ which is continuous yet not continuously differentiable. Then your hypothesis is that for every smooth compactly supported $\varphi$, $x \longmapsto \int_x{\infty}{y\varphi(y)\,dy}-x\int_x{\infty}{\varphi(y)\,dy}$ is continuously differentiable and it is true. But $f$ is not continuously differentiable.

Answer 2

Let $n=1$, and $f(x,y)= 1_{[x, \infty[}(y)$. Tour hypothesis is $\int _x^\infty \phi(y) dy$ is $C^1$ for each $\phi$ , so is satisfied. However $f$ is not continuous.