Can a function have a derivative and not be differentiable?

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 have a 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).

How do you tell if a function does not have a derivative?

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.

What functions are non differentiable?

A function is non-differentiable where it has a “cusp” or a “corner point”. This occurs at a if f'(x) is defined for all x near a (all x in an open interval containing a ) except at a , but limx→a−f'(x)≠limx→a+f'(x) . (Either because they exist but are unequal or because one or both fail to exist.)

What are the examples of non differentiable functions?

Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.

Can a non differentiable function be discontinuous?

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.

Why are discontinuous functions non differentiable?

Take for example the very simple function: f(x)={x+1x≥0,xx<0. 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.

How do you find non-differentiable points?

If f(a-) not equal to f(a+) at x=a, then certainly f(x) is not differentiable at x=a. If f(a-)=f(a+), then check whether f’ (a-) = f’ (a+). If not, then f(x) is not differentiable at x=a.