Question:

Let $f(x)$ be a real function. Number of correct statements among: I) If $f(x)$ is odd on $\mathbb{R}$ and increasing for $x>0$ then it is decreasing for $x<0$. II) If $f(x)$ is odd and $\lim_{x\to 0}f(x)$ exists then $\lim_{x\to 0}f(x)=0$. III) If $f(x)$ is even on $\mathbb{R}$ and known to be increasing for $x>0$, then it is decreasing for $x<0$. IV) If $f(x)$ is even and $f(0)$ exists then $\lim_{x\to 0^+}f(x)=\lim_{x\to 0^-}f(x)$.