If $\lim_{x\to c^-}${$\ln x$} and $\lim_{x\to c^+}${$\ln x$} exists finitely but they are not equal

Question

If $\lim_{x\to c^-}${$\ln x$} and $\lim_{x\to c^+}${$\ln x$} exists finitely but they are not equal (where {} denotes fractional part function),then
$(a)c$ can take only rational values
$(b)c$ can take only irrational values
$(c)c$ can take infinite values in which only one is irrational
$(d)c$ can take infinite values in which only one is rational

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I do not know how to start attempting this question.Its answer given is option $(d)$ Graphing calculator is not allowed.

Answer 1

Hint: The only way that the limits cannot be equal is if $\ln c$ is an integer.

Bigger hint: Hover over the greyed out box to reveal.

$\ln c$ is an integer if and only if $c = e^n$ for some integer $n$, and the only integer $n$ for which $e^n$ is rational is $n=0$.