Mathematics

The key is to examine what happens to $f(x)$ as $x$ approaches 1 from either side. As $x$ gets closer to 1, the denominator $(x-1)^2$ shrinks toward zero, but stays positive, so the fraction $\frac{2}{(x-1)^2}$ grows without bound. Since the function values shoot off to infinity rather than approaching a finite limit or having a removable gap, this is an infinite discontinuity—the function literally cannot be "fixed" by redefining it at a single point.

The key is to examine what happens to $f(x)$ as $x$ approaches 1 from either side. As $x$ gets closer to 1, the denominator $(x-1)^2$ shrinks toward zero, but stays positive, so the fraction $\frac{2}{(x-1)^2}$ grows without bound. Since the function values shoot off to infinity rather than approaching a finite limit or having a removable gap, this is an infinite discontinuity—the function literally cannot be "fixed" by redefining it at a single point.