Value of c=?

$f(x) =\begin{cases} c^2x -c & x \le 1 \\ cx-x & x>1 \end{cases}$

The partial functions of $f(x)$ are continuous for $x<1$ and $x>1$ .(since they are polynomials)

To get $f(x)$ continuous on $(-\infty,\infty\big)$ :

we need $\lim_{x\to 1^-} f(x) = \lim_{x\to 1^+} f(x)=f(1)$ .

This happens only when $c^2-c=c-1\implies c^2-2c+1=0 \implies (c-1)^2=0 \implies c=\boxed{1}$

Placa $x=1$ , $c^2-c=c-1\rightarrow c=1$