Question

If f(x) is continuous on (a,b), is it correct to say it ( f(x) ) has a derivative on (a,b).

Answer 1

no. there are examples of functions that are everywhere continuous but nowhere differentiable.

an easy example to understand is the absolute value function \(f(x) = |x|\) which is continuous for all real x but is not differentiable at x = 0.

Answer 2

thank you @pgpilot326 , is the question implying that (a,b) are two ends of f(x), which are continuous, and one can differentiate a point between a and b?

Answer 3

i'm not sure exactly what you're after with that one...

if f(x) is continuous on (a, b) then it is continuous on that interval. the interval is an open interval with no real endpoints. f(a) nor f(b) is not necessarily defined.

continuity is a necessary condition for differentiability, but not a sufficient one.

what that means is that in order for a function to be differentiable, it must be continuous (at least on some open interval).

however, just because it is continuous on some open interval, that does not mean that it is differentiable there.

Answer 4

ya, I was just trying to say what you have said using "English", and you said it mathematically. It is an open interval. Thank you for making me understand.

Answer 5

yeah, gotta learn to blend the two...

difficult to speak math using only English and difficult to speak English using only math.