if $f$ is continuous and $t \mapsto \partial_t f(x,t)$ is continuous then is $(x,t) \mapsto \partial_t f(x,t)$ continous?

Question

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a continuous function such that $\partial_t f(x,t)$ exists and such that for all $x \in \mathbb{R}$ the function $t \mapsto \partial_t f(x,t)$ is continuous. Then is : $$(x,t) \mapsto \partial_t f(x,t)$$

continuous ?

When doing an exercise on the continuity of integrales with parameters I am using this fact yet I don't know if this is true or false.

It feel like it's false but I wasn't able to find a counter-example.

Thank you !