Is the minimum a continuous function?

Question

Let's consider a continuous function $$ f:\mathbb{R}\times [a,b]\to\mathbb{R}$$ Such that $g(y)=\min_{x\in\mathbb{R}} f(x,y)$ is well defined for every $y\in[a,b]$.

Is it true that $g(y)$ is continuous on [a,b]?

I believe this statement to be true, but I don't know how to prove it. Any hint would be appreciated.

Answer 1

No in general. Counterexample:

$$f : \mathbb R\times [0,1] \to \mathbb R,\ \ \ f(x, y) = \max\left\{\frac{1}{1+(yx)^2},\frac12\right\}. $$

In this example,

$$ g(y) = \begin{cases} 1/2 &\text{if } y>0, \\ 1 & \text{if } y=0.\end{cases}$$