How do you know if a derivative is continuous?

Example of a function that has a continuous derivative: The derivative of f(x) = x2 is f′(x) = 2x (using the power rule). This derivative has met both of the requirements for a continuous derivative: The initial function was differentiable (i.e. we found the derivative, 2x), The linear function f(x) = 2x is continuous.

Are derivative functions continuous?

A differentiable function is necessarily continuous (at every point where it is differentiable). It is continuously differentiable if its derivative is also a continuous function.

Can you find the derivative of a discontinuous function?

It is discontinuous at x=0 (the limit limx→0f(x) does not exist and so does not equal f(0)), but if I find the derivative using the limit above, I get the left and right limits to equal 1. So therefore, the derivative exists.

When is a derivative not continuous?

The derivative, by the product and chain rules, is . That cannot be continuously extended to since there is nothing to control the cosine bit. However, if you resort to the definition of a derivative and determine the derivative at zero directly then you find it does exist (and is ).

Where is a derivative continuous?

How do you find the continuity of a function with two variables?

We define continuity for functions of two variables in a similar way as we did for functions of one variable. Let a function f(x,y) be defined on an open disk B containing the point (x0,y0). f is continuous at (x0,y0) if lim(x,y)→(x0,y0)f(x,y)=f(x0,y0). f is continuous on B if f is continuous at all points in B.

What is a discontinuous derivative?

The basic example of a differentiable function with discontinuous derivative is f(x)={x2sin(1/x)if x≠00if x=0. The differentiation rules show that this function is differentiable away from the origin and the difference quotient can be used to show that it is differentiable at the origin with value f′(0)=0.