Where are functions 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.)

Can a function be not continuous?

In other words, a function is continuous if its graph has no holes or breaks in it. For many functions it’s easy to determine where it won’t be continuous. Functions won’t be continuous where we have things like division by zero or logarithms of zero.

What does it mean if a function is differentiable?

A function is differentiable at a point if it has a derivative there. In other words: The function f is differentiable at x if. lim. h→0.

Where can a derivative not 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. In case 3, there’s a tangent line, but its slope and the derivative are undefined.

Can a function be continuous and not differentiable?

In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.

What 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.

What does non differentiable mean?

Non-differentiable function. In the case of functions of one variable it is a function that does not have a finite derivative. For example, the function is not differentiable at , though it is differentiable at that point from the left and from the right (i.e. it has finite left and right derivatives at that point).

At zero, the function is continuous but not differentiable. If f is differentiable at a point x0, then f must also be continuous at x0. In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable.

How to tell if differentiable?

– Differentiable functions are those functions whose derivatives exist. – If a function is differentiable, then it is continuous. – If a function is continuous, then it is not necessarily differentiable. – The graph of a differentiable function does not have breaks, corners, or cusps.

Is a cusp differentiable?

A function which jumps is not differentiable at the jump nor is one which has a cusp, like |x| has at x = 0. 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.