Continuous Function: Is There an Open Interval Where f is Monotone?

[sbashrawi]

Homework Statement

Let f be continuous on R. Is there an open interval on which f is monotone?

Homework Equations

The Attempt at a Solution

I think there is such interval for non constant function but I am really not sure.

 

[HallsofIvy]

You might want to check out the "Weierstrasse function" which is continuous for all x but differentiable nowhere.

 

[╔(σ_σ)╝]

In general, no.

Continuity doesn't tell you how the fuction approach it's points. If you knew that f '(x) was either negative or positive around some neighbour of your point then you can say if it is monotone or not.