Prove or disprove involving periodic derivatives and functions

[tmlrlz]

Homework Statement

A function f(x) has a periodic derivative. In other words f ' (x + p) = f ' (x) for some real value of p. Is f(x) necessarily periodic? Prove or give a counterexample.

Homework Equations

Periodic functions and Periodic Derivatives

The Attempt at a Solution

To be honest, this question stumped me because the only functions that i can think of when the mention of periodic comes is trig functions. I'm thinking that it must be a trig function which can act as a counterexample, specifically secx,cscx or cotx. however I'm not sure if these functions are periodic in the first place and then if their derivatives are periodic. I'm quite sure their derivatives are periodic but I'm not sure if they are periodic and if they're not, that will act as a counterexample. I'm worried that this might be true, because its much easier to disprove an argument than it is to prove it. If this is true, can someone help me go about the means of proving it. Thank you.

 

[Dick]

The derivative of f(x) can be periodic even if f(x) isn't periodic. Concentrate on finding a counterexample. There's actually another thread on this same problem. The hint given was you know if g(x)=f(x+p)-f(x), so you know g'(x)=0. Does that make g(x) zero?