Let $f$ be a continuous function on $[a,b]$. Show the function $f^*$ defined by $f^*(x) = \sup\left\{ f(y) : a \leq y \leq x\right\}$, for $ x \in [a,b]$, is an increasing continuous function on $[a,b]$
My approach of this is: Since $f$ is continuous, then $f$ is uniformly continuous on $[a,b]$, therefore $$ \forall \epsilon >0, \exists \delta >0,x,y \in [a,b]$$ $$ |x-y| < \delta \implies |f(x) - f(y)|<\epsilon$$ (This is the definition in Ross.) And I would like to use this to show $f^*$ is uniformly continuous on $[a,b]$, but I stuck when I tried to relate the uniform continuity of $f$ and the the definition of $f^*$. I also notice that the definition of $f^*$ implicitly implies that $f(x)\leq f^*(x)$, and there exists a $y_0$ in $[a,x]$ s.t. $f(y_0) = f^*(x)$. Can anyone help me on this?