In any mean value theorem i.e.,rolles ,lagranges & cauchi why the countinuity is verified in [a b] i.e.,in closed interval and why derivability is checked in open interval (a b)???
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Differentiability at the endpoints is not necessary, so there is no need to impose.
any extra conditions here.
Continuity, however is needed at the endpoints, or else Rolle's theorem is false.
Consider the function:
f(x)=x for 1≤x<2 and f(2)=1.
This function is differentiable in (1,2) but continous only in [1,2). The only condition (for Rolle's) that is not satisfied is continuity at x=2, and we can see that the theorem fails in this case.
any extra conditions here.
Continuity, however is needed at the endpoints, or else Rolle's theorem is false.
Consider the function:
f(x)=x for 1≤x<2 and f(2)=1.
This function is differentiable in (1,2) but continous only in [1,2). The only condition (for Rolle's) that is not satisfied is continuity at x=2, and we can see that the theorem fails in this case.
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