Suppose $f:[0,1]^2\to\mathbb{R}$, $(t,x)\mapsto f(t,x)$, is such that for each $t\in[0,1]$ $f(t,\cdot)$ is Lebesgue measurable on $[0,1]$, and for each $x\in[0,1]$ $f(\cdot,x)$ is continuous everywhere on $[0,1]\ni t$.

**1.** Does this imply that $f(t,x)$ is measurable on $[0,1]^2$?

**2.** Does this imply that the function $g(x)=\min\limits_{t\in[0,1]}f(t,x)$ is measurable on $[0,1]$?