sequence of continuous functions converging pointwise to reimann integrable funtion

Question

$$a,b\in \mathbb{R}.\, \, f:[a,b]\rightarrow \mathbb{R}\textrm{ is bounded. } f:[a,b]\rightarrow \mathbb{R}\textrm{ is reimann integrable .}\\ \textrm{Prove that }\exists \left \{ f_{n} \right \}_{n=1}^{\infty }\textrm{ such that }f_{n}:[a,b]\rightarrow \mathbb{R}\textrm{ is continuous }\forall n\in \mathbb{Z}^{+} \textrm{ such that }\\ \lim_{n\rightarrow \infty }f_{n}(x)=f(x)\: \: \forall x\in [a,b]$$ I know this is true when f is continuous. What about when f has at most countable number of discontinuities