"Let r be a rational number greater than 1. Let f:R->R be a function that satisfies the condition that for all real numbers x and y,(adsbygoogle = window.adsbygoogle || []).push({});

[tex]|f(x)-f(y)|\leq (x-y)^r[/tex]

Prove that f is a constant."

Perhaps there's some subtlety which I misunderstood, but even a constant function fails to satisfy the conditions for some choice of r. For instance, let r=3 and f be a constant. Let x<y, and the inequality fails, i.e.,

[tex]|f(x)-f(y)|=0\leq (x-y)^3 <0[/tex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Constant Function

**Physics Forums | Science Articles, Homework Help, Discussion**