Is function $f$ continuous? $$f(x) = \begin{cases} x-1.5 & \text{ if } x<1\\\\ \dfrac{1-x}{x^2-1} & \text{ if } x>1\end{cases}$$
Is this function continuous or not? Some people say it is because the value $x=1$ is not defined for the function while others say it is not continuous because it has a hole at $x=1$. Which is correct?

The function, as it stands now with definitions for $x<1$ and $x>1$ and neither $x\leq1$ nor $x\geq1$, is undefined at $x=1$. Thus it is neither continuous nor discontinuous for $x=1$; it simply isn't. At all other points, it is continuous.
It is also worth noting that if you fill in the gap with $f(1)=-0.5$, then the function becomes defined and continuous on the whole number line.
