Continuity of the minimizer of a parametrized function

Question

Let $f: \mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}$ be a smooth function and let $g : \mathbb{R}^2 \to \mathbb{R}^+$ be continuous. Define $$ M(a,b) := \min_{x \in [0,g(a,b)]} f(a,b,x), $$ that means $M(a,b)$ is the minimum of $f(a,b,\cdot)$ on the interval $[0,g(a,b)]$. Do the above assumptions on $g$ and $f$ suffice so conclude that $M$ is a continuous function? If not, can we make stronger assumptions which imply continuity of $M$?