A Function with no Derivative?

[FeDeX_LaTeX]

Hi,

Is there any function defined such that it is non-differentiable at every point?

Of course, cusps and asymptotic graphs aren't differentiable at those points, but what about one that can't be differentiated anywhere? I know there are crazy functions like defining some function to be 0 if x is rational and 1 if x is irrational.

Thanks.

 

[Office_Shredder]

There are in fact functions which are continuous at every point and differentiable nowhere. They're pretty complicated. One example:
http://en.wikipedia.org/wiki/Weierstrass_function

One example that's maybe easier to explain intuitively is Brownian motion. The basic idea is that you have an object that's moving up or down at random. Then f(t) is the height of the object at time t. Then the height of the object is continuous, but over arbitrarily small time intervals it changes whether it's moving up or down, so the derivative is not defined

 

[FeDeX_LaTeX]

Thanks for the reply. I have heard of the Weierstrass function but did not know it was differentiable nowhere. Thanks for this.

I sort of get what you mean by the Brownian motion example.

How would one plot a graph of the function I alluded to, where f(x) = 0 if x is rational and f(x) = 1 if x is irrational?

 

[Number Nine]

How would one plot a graph of the function I alluded to, where f(x) = 0 if x is rational and f(x) = 1 if x is irrational?

You wouldn't. The best you could do is plot a grossly simplified kind-of sort-of approximation with a discrete number of values on a small interval.

 

[CellCoree]

I can confirm that there are indeed functions that are non-differentiable at every point. These functions are called "nowhere differentiable" or "pathological" functions. They are often constructed through complex mathematical techniques and do not have a derivative at any point. One example is the Weierstrass function, which is defined as the sum of an infinite series of cosine functions with increasing frequencies and decreasing amplitudes. This function is continuous but not differentiable at any point. Another example is the Blancmange function, which is defined as the sum of a sequence of triangular waves. It is continuous but not differentiable at any point. These functions may seem strange and counterintuitive, but they have important applications in fractal geometry and chaos theory. So, to answer your question, yes, there are functions that cannot be differentiated anywhere, and they are a fascinating subject of study in mathematics and science.