How to check the continuity of this function defined as follows 
The function $f:\Bbb R\to\Bbb R$ defined by $f(x)=\min(3x^3+x,|x|)$ is
(A) continuous on $\Bbb R$, but not differentiable at $x=0$.
(B) differentiable on $\Bbb R$, but $f\,'$ is discontinuous at $x=0$.
(C) differentiable on $\Bbb R$, and $f\,'$ is continuous on $\Bbb R$.
(D) differentiable to any order on $\Bbb R$.

My attempt: Here,$f(x)=|x|,x>0$;$f(x)=3x^3+x,x<0$;$f(x)=0$ at $x=0.$ Also,$Lf'(0)=Rf'(0)=1.$ So,$f$ is differentiable at $x=0.$ But I am having trouble to check whether $f'$ is continuous at $x=0$ or not. Can someone point me in the right direction?Thanks in advance for your time.

HINT: If $x<0$, $f\,'(x)=9x^2+1$, so $$\lim_{x\to 0^-}f\,'(x)=\lim_{x\to 0^-}\left(9x^2+1\right)=1\;.$$
Is this the same as $\lim_{x\to 0^+}f\,'(x)$? Are both one-sided limits equal to $f\,'(0)$?
