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?
Continuity is a property of a function where the output values change smoothly as the input values change. In other words, there are no sudden jumps or breaks in the graph of the function.
A derivative is a measure of how a function changes as its input values change. It represents the slope of the tangent line at a specific point on the graph of the function.
The main difference between continuity and derivatives is that continuity is a property of the function as a whole, while a derivative is a property of a specific point on the function.
Continuity is a necessary condition for a function to have a derivative at a specific point. In other words, if a function is not continuous at a certain point, it cannot have a derivative at that point.
Yes, it is possible for a function to be continuous at a certain point but not have a derivative at that point. This usually occurs when the function has a sharp turn or corner at that point, which makes it impossible for a tangent line to be drawn.