For what values of $c$ is the function continuous everywhere?

$f(x)=\{\begin{array}{ccc}cx& 2,& x<mn>0</mn>\\ x^2-3,& x0\end{array}$

Explain whether the function $f(x)=\frac{x1}{x-1}$ is continuous at $x=1.$ Justify your answer.

Bonus Question $(2$ points$)$

True or False: If $f(x)$ is continuous at $x=c,$ then $\underset{xc}{lim}f(x)$ always exists and is equal to $f(c).$ If true, justify. If false, provide a counterexample.