$\text{ Let }f\left(x\right)=\left\{\begin{array}{l}\frac{g\left(x\right)-g\left(a\right)}{x-a},x\le a\\ g^{\mathrm{\prime }}\left(a\right),x=a\end{array}\text{,where }g\text{ is a function derivable at }x=a\right.\text{, then at x=a}$

• A

  $f\text{ is continuous }$

• B

  $f\text{ is discontinuous }$

• C

  $f\text{ is continuous only if }{g}^{\mathrm{\prime }}\left(a\right)=0$

• D

  $\text{None of these}$