Thursday, 31 January 2013

real analysis - approximating a riemann integrable function by sequences of step functions and sequences of continuously differentiable functions

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.

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