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 second derivative exist if First derivative does not?

The second derivative is the derivative of the first derivative of the function. If the first derivative of a function does not exist, then you cannot find a derivative of a non-existent function to obtain a second derivative. Thus, the answer is no.

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

Hence, differentiability is when the slope of the tangent line equals the limit of the function at a given point. This directly suggests that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well.

Can a discontinuous function be differentiable?

If a function is discontinuous, automatically, it’s not differentiable.

What does the first derivative tell you about concavity?

When the function y = f (x) is concave up, the graph of its derivative y = f ‘(x) is increasing. When the function y = f (x) is concave down, the graph of its derivative y = f ‘(x) is decreasing.

What is an example of a differentiable function with discontinuous derivative?

The basic example of a differentiable function with discontinuous derivative is f (x) = { x 2 sin (1 / x) if x ≠ 0 0 if 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.

Can a periodic function have a derivative with removable discontinuities?

(More precisely, a periodic function whose left- and right-hand limits are unequal at some points can, nonetheless, have derivative with removable discontinuities.) The trig functions and the principal branches of their inverses are good at creating this type of behavior.

Can two different functions have the same derivative?

Yes, two different functions can have the same derivative under certain conditions. The reasoning is as follows. Consider two functions φ (x) and ψ (x) which are continuous and differentiable at all points x in the interval a < x < b. We know that derivative of a function at a given point is the slope of the graph of that function at that point.

How do you prove a function is differentiable at the origin?

( 1 / x) if x ≠ 0 0 if 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. A graph is illuminating as well as it shows how ± x 2 forms an envelope for the function forcing differentiablity.