Prove that given any Riemann Integrable function $f$ on $[a,b]$, and given any $\varepsilon>0$ one can find a continuous function $g$ on $[a,b]$ such that $$\int_a^b|f(x)-g(x)|dx<\varepsilon $$
I could not even start this problem, except having some stray ideas. Some thoughts are using Step functions but they are piecewise continuous only.
Please give me some hints only on how to construct such a $g$. Also, please note that I wish to solve this WITHOUT using Weierstrauss Approximating Polynomials.