Prove that there exists a continuous function $g$ such that $\int_a^b|f-g| \lt \epsilon$

Question

Assume that $f$ is Riemann-integrable on the interval $[a,b]$ and $\epsilon$ is a positive number.

Prove that there exists a continuous function $g$ such that $\int_a^b|f-g| \lt \epsilon$ .

Note : The problem is the existence of the function. I don't know how to show it. Any hint or clue would be useful. An answer would be great.

Thanks in advance.