# Continous function

1. Sep 27, 2004

### chjopl

continuous function

Is there a continuous function that has no tangent line at all? If so what is it? I know it must be made up of cusps and corners

Last edited: Sep 28, 2004
2. Sep 28, 2004

### BLaH!

A function that doesn't have a tangent at a point means that the function's derivative doesn't exist at that point. You're right, functions are non-differentiable at cusps or corners. Examples of functions that don't have derivatives at one or more points include the absolute value function: $$y = \left|x\right|$$ or the Heaviside step function: $$\theta(x) = \left\{\begin{array}{cc}0,&\mbox{ if } x\leq 0\\1, & \mbox{ if } x>0\end{array}\right.$$

3. Sep 28, 2004

### phoenixthoth

4. Sep 28, 2004

### chjopl

That cleared it up but i couldnt figure out the equation of the function.

5. Sep 28, 2004

### arildno

To give you another one:
Define f(x) as:
$$f(x)=\sum_{n=0}^{\infty}\frac{\sin((n!)^{2}x)}{n!}$$
f'(x) cannot be defined at any point, although f(x) is continuous for all x.
This is, I believe, Weierstrass' first published example of such a function.

6. Sep 28, 2004

### chjopl

What does the n! stand for

7. Sep 28, 2004

### Pyrrhus

It means Factorial, look it up.