It's true that if f is differentiable in (a,b) then f is continuous in (a,b).
But would it be correct to say that if f is differentiable in (a,b) then f is continuous in [a,b] ?
MIT 18.01 Single Variable Calculus (OCW)
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I don't think it's necessarily true. I mean, it could be true, but could also be false.
All differentiable functions are continuous on that same interval.SO THE FIRST STATEMENT IS TRUE. but
NOT all continuous functions are differentiable .SO FOR THE SECOND STATEMENT I THINK IT IS FALSE.
Pasta is right. Just consider most rational power functions and the absolute value function.
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If f(x) is differentiable at (a,b), then it has to be continuous the the same point.
But, it is not necessary that every continuous function is differentiable.
f(x) = |x|
Hence we conclude that
1) Differentiability implies continuity
2) Continuity does not imply differentiability.