Let f be the function defined by $f(x)=\left\{\begin{matrix}\frac{x^2-1}{x^2-2|x-1|-1}&;&x≠1\\1/2&;&x=1\end{matrix}\right.$

the function is not continuous at x = 1

For x < 1, $f(x)=\frac{x^2-1}{x^2+2x-3}=\frac{x+1}{x+3}$;  $∴\underset{x→1^-}{\lim}f(x)=\frac{1}{2}$

For x > 1, $f(x)=\frac{x^2-1}{x^2-2x+1}=\frac{x+1}{x-1}$

$∴\underset{x→1^+}{\lim}f(x)=∞$

∴ The function is not continuous at x = 1