Supremum of a Continuous Function is Continuous

I'm working on this problem from Elementary Analysis by Ross which is intuitive when sketched but keeps stymieing me when I try to write it out.

Let $f$ be a continuous function on $[a,b] \subset \mathbb{R}$. Define $f^\star (x)$ as:

$$f^\star(x) = \sup \{f(y)\mid a \leq y \leq x \}$$Prove that $f^\star$ is a continuous increasing function on $[a,b]$.

Things I've figured out: (these are mostly trivial)

• $f$ is uniformly continuous on $[a,b]$.
• Since $[a,b]$ is closed and bounded there exists some $c \in [a,b]$ such that $f(c) \geq f(x)\ \forall x \in [a,b]$. (In words: the supremum is actually attained.)
• If $c \in [a,b]$ is as above, then $f^\star$ is constant (and hence continuous) on $[c,b]$ (which is possibly a singleton).

This seems completely obvious when you actually draw a continuous function but translating that to a formal proof eludes me...