Non-differentiable curves (1 Viewer)

[hs17]

Can people list some curves that aren't differentiable?

 

[Qeru]

Can people list some curves that aren't differentiable?

A curve isn't differentiable at a point if it satisfies either of two conditions:
1. Its not continuous at that point (i.e. its not defined there)
2. Has a cusp (a steep point in a graph)
A classic example of a curve with a cusp is which isn't differentiable at x=0. Any curve having a discountinity isn't differentiable at that point. Now if you want a function that isn't differentiable anywhere yet is continuous: https://en.wikipedia.org/wiki/Weierstrass_function

 

[CM_Tutor]

A curve isn't differentiable at a point if it satisfies either of two conditions:
1. Its not continuous at that point (i.e. its not defined there)
2. Has a cusp (a steep point in a graph)
A classic example of a curve with a cusp is which isn't differentiable at x=0. Any curve having a discountinity isn't differentiable at that point. Now if you want a function that isn't differentiable anywhere yet is continuous: https://en.wikipedia.org/wiki/Weierstrass_function

One quibble... "not continuous" at a point is not the same as "not defined" at that point. A curve is continuous at a point if the value of the function at that point is the same as the limits of the function as it approaches from either side.

Consider a function like



which looks like the line but with the point missing, replaced by the point . It is defined at as . It is not continuous at , however, as approaching from either above or below, .

It is possible to have a function that is not continuous at any point in its domain. The classic example is the Dirichlet function,



which takes the value 1 if is rational and 0 if is irrational.