Suppose that $f$ is Riemann integrable on $[0,M]$.
How can I show that a) $f$ can be approximated uniformly by a sequence of finite step functions? and b) by a sequence of continuously differentiable functions?
Any hints on how to handle this?
Well I thought that for a), since $f$ is Riemann integrable then there is a partition such that the sum of the product of the oscillation at each partition and the length of the partition is uniformly small. But the oscillation at each partition is a difference of two step functions, the one which bounds the function above and the one which bounds the function below... Help would be appreciated.