I got this statement that I am trying to prove
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that for all $x\neq 0$, $f'(x)>0$ and for $x=0$, $f'(x)=0$, how do I prove that $f$ is increasing on $\mathbb{R}$ by the definition of increasing function.
My try: I showed that $f$ is increasing on $\mathbb{R}-\{0\}$ using the theorem on increasing functions and their derivatives, but at $x=0$ I came into difficulty and wasn't able to proceed that much (do I miss something, maybe MVT or Rolle's theorem ).
Some hints will be appreciated.