Here are pictures of parts of the graph of f(x)=x+2x² sin(1/x). This function is everywhere differentiable. Its derivative at x=0 is f '(0)= 1, but arbitrarily close to x=0 you can find pieces of the graph of f which are increasing and other pieces which are decreasing: So: Even if f '(0)>0, the function f does not have to be increasing near x=0.