hello (pardon me if this is a lame question, but i got to still ask) If a function is uniformly continuous (on a given interval) then is it required for the derivative of the function to be continuous? I was thinking as per the definition of Uniform continuity, f(x) should be as close to f(y) as possible, for x around y, meaning change in f should not be sudden at some point within the interval; it should rise or fall in a uniform manner, suggesting the slope of the function at different points should change gradually and be real always (i.e. never infinite), that implies the derivative should be continuous in the open interval. So if we have a function whose derivative is continuous then can we say it is uniformly continuous on the closed interval? (I must mention i won't be able to understand very rigorous proofs, so if possible explain this either by counter example, if any, or geometrically. ) thank you..