Continuous Function: No Tangent Line?

[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

 

[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: $$\theta(x) = \left\{\begin{array}{cc}0,&\mbox{ if } x\leq 0\\1, & \mbox{ if } x>0\end{array}\right.$$

 

[phoenixthoth]

A function can be nowhere differentiable yet everywhere continuous. It's hard to draw but it does exist. Here is an example

http://www.math.tamu.edu/~tom.vogel/gallery/node7.html

 

[chjopl]

A function can be nowhere differentiable yet everywhere continuous. It's hard to draw but it does exist. Here is an example

http://www.math.tamu.edu/~tom.vogel/gallery/node7.html

That cleared it up but i couldn't figure out the equation of the function.

 

[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.

 

[chjopl]

What does the n! stand for

 

[Pyrrhus]

What does the n! stand for

It means Factorial, look it up.