Question

Options related to the function f:

a. f is continuous everywhere but not differentiable exactly at one point.
b. f is differentiable everywhere.
c. f is not continuous exactly at two points.
d. f is continuous everywhere but not differentiable exactly at two points.

Answer 1

The function f is continuous everywhere but not differentiable exactly at one point. Therefore, option (a) is the correct choice.

Step-by-step explanation:

The correct option related to the function f is:

Option (a) f is continuous everywhere but not differentiable exactly at one point.

To be continuous everywhere, the function must have no jumps or holes in its graph. However, it can still have a non-differentiable point, where the derivative does not exist. This can be in the form of a sharp corner or a vertical tangent.

For example, the absolute value function f(x) = |x| is continuous everywhere but not differentiable at x = 0, where it has a sharp corner.