TITLE: If $f,g: [a,b] \to \mathbb{R}$ are bounded with $g$ continuous and $f>g$, is there a continuous $h:[a,b] \to \mathbb{R}$ with $f>h>g$?
QUESTION [6 upvotes]: Is the following claim true:

Let $f,g :[a,b]  \to  \mathbb{R}$ be bounded functions with $g$ continuous and $f>g$. Then there exists a continuous function $h: [a,b]  \to  \mathbb{R}$, such that $f>h>g$.

I am only interested in whether or not the claim is true or not, so please leave the proof to me if the claim is true. However, if the claim is false, I welcome counter examples!
Any help is greatly appreciated!

REPLY [4 votes]: You can take a similar example as in your other question: