Continuity of $f(x)$ on $[a,b]$ implies continuity of $f(t) = \displaystyle \min_{a \leq x \leq t}f(x)$ at [a,b].

Question

Let $f(x)$ be a continuous function on $[a,b]$.

Show that $F(t) = \displaystyle \min_{a\leq x\leq t}f(x)$ is also continuous on $[a,b]$.

Statement seems obvious for me, especially when I try to draw some special cases of this problem, but I fail to derive a rigorous proof.