If $f$ is Lipschitz of order $\alpha > 1$ over $[a,b]$, prove $f$ is constant

Question

If $f$ is Lipschitz of order $\alpha > 1$ over $[a,b]$, prove $f$ is constant.

We are given that for all $x,y \in [a,b]$ there exists $C>0$ such that $|f(x)-f(y)| \leq C|x-y|^{\alpha}$. We would like to show that $f(x)-f(y) = 0$ in this domain, but I don't see how to get to that conclusion since $C>0$.

Also, does the question mean $f$ is constant over $[a,b]$ or everywhere?

@Fnacool We know that $f(x)$ is continuous over the interval, but how do I show differentiability? — user19405892, Mar 22 '16 at 03:02
Once you know it's constant on $(a,b)$, you can show it's constant on $[a,b]$ by continuity. — , Mar 22 '16 at 03:05

score 3 · Accepted Answer · answered Mar 22 '16 at 03:03

3

It may help if you notice that $|f(x)-f(y)| \leq C|x-y|^{\alpha}$ is equivalent to $\frac{|f(x)-f(y)|}{|x-y|} \leq C|x-y|^{\alpha-1}$, and $\alpha -1 > 0$ by assumption.

answered Mar 22 '16 at 03:03

siegehalver

1,296

I still don't see how this shows the derivative is $0$. – user19405892 Mar 22 '16 at 03:09
So, if you take a limit as $y \to x$, the left hand side is the definition of derivative, and the right hand side will go to zero noting that $\alpha-1>0$ – siegehalver Mar 22 '16 at 03:11

score 3 · Answer 2 · answered Mar 22 '16 at 04:26

3

Without derivatives (just for fun): It suffices to show $f(b)=f(a).$ Let $a=x_0 < x_1 < \cdots < x_n$ be the uniform partition of $[a,b]$ into $n$ subintervals of length $(b-a)/n.$ Then

$$|f(b)-f(a)| = |\sum_{k=1}^{n}(f(x_k) - f(x_{k-1}))| \le \sum_{k=1}^{n}|f(x_k) - f(x_{k-1})|$$ $$\le \sum_{k=1}^{n}C(x_k-x_{k-1})^\alpha = nC[(b-a)/n]^\alpha = C(b-a)^\alpha n^{1-\alpha}.$$

Because $1-\alpha < 0,$ $n^{1-\alpha}\to 0$ as $n\to \infty.$ Since the above holds for any $n,$ we have $f(b)-f(a) = 0$ as desired.

answered Mar 22 '16 at 04:26

zhw.

107,943

let me ask you one question (just for fun). suppose instead of closed interval domain is just a subset of real number and function is Lipschitz of order greater than one. Will function be constant? if not then exactly for what type of domain it will constant. – CHOUDHARY bhim sen Feb 16 '25 at 17:12

If $f$ is Lipschitz of order $\alpha > 1$ over $[a,b]$, prove $f$ is constant

2 Answers2