If f is continuous at x = c, then f is differentiable at x = c ?
We will use the concept of limit, continuity, and differentiability to find the required reasoning.
Answer: It is not necessary that if a function f is continuous at x = c, then f will be differentiable at x = c.
Let us see how we will use the concept of limit, continuity, and differentiability to find the required reasoning.
Let us consider an example of f(x) = | x |. We can easily see that the function | x | is continuous at point x = 0 but is not differentiable at x = 0 because of more than one slopes at x = 0 . That is the curve has a sharp curve at x = 0.
In the above image, we can clearly see that |x| has more than one slope at x = 0, which means when we will differentiate it then we must get more than one slope. But this is not possible, because, at x = 0 we can draw an infinite number of tangents because the graph takes a sharp turn at x = 0. Hence, the graph is not differentiable at x = 0.