0

When $c=1$ we know that it is Lipschitz, hence the function $f(x)$ is uniformly continuous. But if $c>1$ then what are the possibilities may come out?

Note- Function is defined on real numbers.

Kushal Bhuyan
  • 7,110

2 Answers2

3

The function is constant: take $y=x+h$, then $$ \lvert f(x+h)-f(x) \rvert \leqslant h^c, $$ and dividing both sides by $h$ and taking the limit as $h \to 0$ shows that $f'(x)=0$. But this is true for any $x$, and the only continuous functions with zero derivative everywhere are constant.

Chappers
  • 67,606
2

These functions are called Hölder functions. If the exponent is $> 1$ and the space isn't too weird, only constant functions satisfy the condition.

Hint of proof: if $f(x) \neq f(y)$, you can find a chain of small intervals which connect $x$ to $y$ (if the distance between $x$ and $y$ is $d$, you will roughly need $n$ intervals of size $d/n$). But the Hölder condition implies that these intervals will be sent to intervals of size $\leq (d/n)^c$, do the distance between $f(x)$ and $f(y)$ has to be $\leq n \times \left(\dfrac dn\right)^c$. As this quantity converges to $0$ when $n$ grows, you have a contradiction.

PseudoNeo
  • 9,709
  • I didnot know about Holder functions. Thanks. +1 – Kushal Bhuyan Oct 12 '15 at 17:06
  • For $c < 1$, they are a really important class of functions... – PseudoNeo Oct 12 '15 at 17:07
  • I am new to this class of functions. So what they really serve beyond the normal functions? – Kushal Bhuyan Oct 12 '15 at 17:10
  • Well, they are not as important as, for example $C^k$ functions, but you see them a lot in certain parts of mathematics. For example, the proof I gave is basically a particular case of the fact that Hölder maps interact particularly well with some notions of fractal dimensions. There are also some parts of mathematics where interesting phenomena happen for functions that are smoother than $C^0$ but less smooth than $C^1$. To explore this gap, the "scale" of Hölder maps of exponent $\alpha \in (0,1]$ is a very natural probe. But you shouldn't worry about them just yet :-) – PseudoNeo Oct 12 '15 at 17:19
  • Glad to know something new :-)..cheeers – Kushal Bhuyan Oct 12 '15 at 17:23