Does the derivative of a function at a point of discontinuity exist?

The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s no derivative — that happens in cases 1 and 2 below. An infinite discontinuity like at x = 3 on function p in the above figure.

Can a function be discontinuous at a point but differentiable at that same point?

The converse of the differentiability theorem is not true. 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 discontinuous function have a limit at the discontinuous point?

No, a function can be discontinuous and have a limit. The limit is precisely the continuation that can make it continuous. Let f(x)=1 for x=0,f(x)=0 for x≠0.

How can you tell if a function is differentiable at a point?

A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean? It means that a function is differentiable everywhere its derivative is defined. So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable.

Can function be discontinuous and differentiable?

you can not differentiate discontinuous functions because the first rule of differentiation is that a function must be continuous in its domain to be a differentiable function.

Can discontinuous be differentiable?

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

Can a function be discontinuous?

Discontinuous functions are functions that are not a continuous curve – there is a hole or jump in the graph. It is an area where the graph cannot continue without being transported somewhere else.

Is there a limit at a point discontinuity?

A function being continuous at a point means that the two-sided limit at that point exists and is equal to the function’s value. Point/removable discontinuity is when the two-sided limit exists, but isn’t equal to the function’s value.