Tuesday, 11 November 2014

real analysis - Prove that a function whose derivative is bounded is uniformly continuous.



Suppose that f is a real-valued function on R whose derivative exists at each point and is bounded. Prove that f is uniformly continuous.


Answer



Since f is bounded then there's M>0 s.t.
|f(x)|MxR

hence by mean value theorem we find
|f(x)f(y)|M|xy|x,yR
so f is a lipschitzian function on R and therefore it's s uniformly continuous on R.


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