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 the function is continuous at x = c, then it 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 sharp curves at x = 0.
In the above image, we can clearly see that |x| has two slopes at x = 0, which means when we will differentiate it then we must get two slopes. 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 derivable at x = 0.