Needed clarification regarding Rolles theorem

[seshikanth]

Hi,
If we consider Rolle's Theorem:
"If [1] f is continuous on [a, b],
[2] differentiable in (a,b), and
[3]f (a) = f (b), then there exists a point c in (a, b) where f'(c) = 0."

In the above theorem, i did not get why condition [1] is needed as we have differentiability => continuity. Can Holes at a and b (removable discontinuity) can be taken care by condition [3]?

Thanks,

 

[l'Hôpital]

Try piecewise things. For example,
what about f(x) = 1/x for x in (0,1] and 1 for x = 0? Then "f(0)" is defined (namely, f(0) = 1) and we have f(0) = f(1) but obviously f'(c) is never zero on (0,1)

 

[HallsofIvy]

Notice that the intervals are different. f is assumed to be differentiable on the open interval (a, b) so it is continuous on that interval but not necessarily on the closed interval [a, b] as l'Hôpital's example shows.