Function Monotone on Some Interval

[Icebreaker]

I need the following proposition in order to prove another theorem, and I can't seen to find it in my textbook. Any hints on how to proceed, or whether it's actually TRUE, would be helpful.

"If f is defined and continuous on some interval I, then there exists subintervals I'=[x-a,x+b], for some real numbers a and b, at every point x in I such that f is monotone on I'."

 

[StatusX]

I don't think that's true. Consider the function:

$$f(x)=\left\{\begin{array}{cc}x \mbox{ sin}(\frac{1}{x}),&\mbox{ if } x \neq 0\\0,&\mbox{ if } x=0\end{array}$$

This is continuous, but is not monotone on any interval containing 0.

 

[Icebreaker]

Ok, I've changed my approach. What about:

"If f is defined and continuous on an interval I and c is in I such that f(c)>0, then there exists an interval I' in I where c is in I' such that f(x)>0 for every x in I'."

 

[StatusX]

That's true. Just use the epsilon delta definition of continuity, taking delta as f(c).