Checking the three-part continuity test at x = 1

First check part 1: is $c(1)$ defined? Yes, we are told $c(1) = 0$, so this part of the definition is satisfied. Next check part 2: does $\lim_{x\to 1} c(x)$ exist? Since $c(x)$ is only defined for $x>1$, the two-sided limit equals the right-hand limit, $\lim_{x\to 1^+} \ln(x-1)$.

As $x \to 1^+$, the quantity $x - 1 \to 0^+$, and since $\ln(u) \to -\infty$ as $u \to 0^+$, this one-sided limit does not exist (it diverges to $-\infty$). Because the limit fails to exist, part 2 of the continuity definition fails, so $c(x)$ is not continuous at $x=1$. Since the failure is due to a one-sided limit being infinite (a vertical asymptote), this is an infinite discontinuity, not a removable or jump discontinuity.