The discussion centers around the concepts of continuity and differentiability in mathematics, exploring their definitions, relationships, and the conditions under which a function can be continuous but not differentiable. Participants examine the limits involved in both concepts and provide examples to illustrate their points.
Participants generally agree on the definitions of continuity and differentiability, but there remains some confusion and debate regarding their relationship, particularly in cases where a function is continuous but not differentiable.
The discussion highlights the importance of understanding the specific limits involved in the definitions of continuity and differentiability, as well as the implications of these definitions for different types of functions.
What "limit exists"? The limit you look at to determine if f(x) is continuous at x= a, is [itex]\lim_{x\to a} f(x)[/itex] while the limit you look at to determine if f(x) is differentiable at x= a is [itex]\lim_{h\to a} (f(a+h)- f(a))/h[/itex]. It is easy to show that if a function is differentiable at x= a, it must be continuous but the other way is not true.cjaylee said:Hey. I am quite confused by continuity and derivatives. Both are finding the limits of a particular function as x approaches a. Then why is it that a graph that is continuous cannot be differentiable? If it is continuous, it means that the limit exists and so, it should be differentiable right?