Continuous/analytic functions

1. May 8, 2011

lavster

if a function is continuous, does this mean that it is analytic. And if a function is analytic does this mean it is continuous?

thanks

2. May 8, 2011

gb7nash

No. For example, f(x) = |x| is not differentiable at x = 0. However, it is continuous.

Yes. Every analytic function has the property of being infinitely differentiable. Since the derivative is defined and continuous, the function is continuous everywhere.

Last edited: May 8, 2011
3. May 8, 2011

lavster

exellent -cheers :)

4. May 8, 2011

Studiot

An analytic function is a function that can can be represented as a power series polynomial (either real or complex).

That is it posesseses a Taylor/Mclaurin expansion.

5. May 8, 2011

Gib Z

Not exactly true. A function is analytic at a point $z_0$ if it is smooth (infinitely differentiable) there, and it's Taylor Series centered at $z_0$ converges to the function on some open set containing $z_0$.

Merely being smooth is not enough - For example
$$f(x)=\begin{cases}\exp(-1/x) \mbox{ if } x> 0 \\ 0 \mbox{ if }x\le0,\end{cases}$$

This function is smooth at 0, with all its derivatives there being 0. Thus, it has a Taylor Series expansion at x=0, $$\sum_{n=0}^{\infty} \frac{0}{n!} x^n = 0$$, but that does not coincide with the value of the function for any positive x, so f(x) is smooth (and has a Taylor Expansion), but is not analytic.