Question

Can a function be differentiable but not continuous.

Answer 1

A function cannot be differentiable but not continuous. The function must be continuous at a point for it to be differentiable there, and its first derivative must also be continuous unless there are specific exceptions.

Step-by-step explanation:

No, a function cannot be differentiable but not continuous. In mathematics, particularly in calculus, the concept of differentiability and continuity are closely related. For a function to be differentiable at a certain point, it must first be continuous at that point. This means if you have a function y(x), it must be a continuous function if you intend to differentiate it at any point x.

Furthermore, the first derivative of the function with respect to x, denoted as dy(x)/dx, must also be continuous. There are some exceptions, such as when dealing with potential functions where V(x) might approach infinity, but in standard real-valued functions, differentiability implies continuity. A function that has discontinuities cannot have a derivative at those points of discontinuity because the behavior of the function is not 'smooth' there. In the context of probability, we often deal with continuous probability density functions, which are functions f(x) where the area under the curve and above the x-axis represents a probability, thus implicitly requiring the function to be continuous.

Answer 2

We see that if a function is differentiable at a point, then it must be continuous at that point. There are connections between continuity and differentiability. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on

Step-by-step explanation: