A standard result in introductory calculus classes is that, if a function has positive derivative on an open interval, then it's increasing there.
Based on this, students tend to think that, if $f'(a)>0$, then $f$ must be increasing "near $a$."
However, the example $f(x) = 2x^2\sin(1/x)+x$ (set $f(0)=0$) shows that this is quite false!
