Can a continuous function have a discontinuous derivative?

The differentiability theorem states that continuous partial derivatives are sufficient for a function to be differentiable. It is possible for a differentiable function to have discontinuous partial derivatives. An example of such a strange function is f(x,y)={(x2+y2)sin(1√x2+y2) if (x,y)≠(0,0)0 if (x,y)=(0,0).

Can a function have 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.

What is an example of a function that is continuous but not differentiable?

In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.

Does the derivative of a continuous function have to be continuous?

No. piecewise continuous functions need not have continuous derivatives.

What makes a derivative discontinuous?

The derivative of a function at a given point is the slope of the tangent line at that point. A removable discontinuity — that’s a fancy term for a hole — like the holes in functions r and s in the above figure. An infinite discontinuity like at x = 3 on function p in the above figure.

What is continuous derivative?

A function is said to be continuously differentiable if the derivative exists and is itself a continuous function. Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.

What is derivative discontinuity?

The derivative discontinuity is a key concept in electronic structure theory in general and density functional theory in particular. The electronic energy of a quantum system exhibits derivative discontinuities with respect to different degrees of freedom that are a consequence of the integer nature of electrons.

Is the Cantor function continuous?

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1.

Can a derivative be continuous?

Simply put, differentiable means the derivative exists at every point in its domain. Consequently, the only way for the derivative to exist is if the function also exists (i.e., is continuous) on its domain. Thus, a differentiable function is also a continuous function.

What are some examples of continuous functions with discontinuous derivatives?

The simplest examples of continuous functions, with discontinuous derivatives in some point, are usually of the form: ( 1 / x) if x ≠ 0 0 if x = 0. ( 1 x) does not exist for x → 0.

What is an everywhere differentiable function that is as discontinuous as possible?

This allows us to construct an everywhere differentiable function with derivative that is discontinuous on the union of those Cantor sets, which is a set of full measure. I guess that you are looking for a continuous function f: R → R such that f is differentiable everywhere but f ′ is ‘as discontinuous as possible’.

Is x = 1 a discontinuous graph?

, a discontinuous graph. We observe that a small change in x near displaystyle {x}= {1} x = 1 gives a very large change in the value of the function. For a function to be continuous at a point, the function must exist at the point and any small change in x produces only a small change in

How do you know if a function is continuous?

For a function to be continuous at a point, the function must exist at the point and any small change in x produces only a small change in `f (x)`. In simple English: The graph of a continuous function can be drawn without lifting the pencil from the paper.