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.
Best Answer
Use $\mathrm{osc}(f, I) = \sup(f,I) - \inf(f,I)$ for each interval of your partition.
Edit: You can use either the upper step function or lower step function to approximate $f$. If $L$ denotes the lower step function (defined by picking the infimum on each partition interval), why is $|f - L| = |f - \inf_I(f)|$ small for each interval $I$?