I have the following function sequence:
$$ f_n(x) = \begin{cases} -1 & -1 \leq x \leq -\frac{1}{n} \\ nx & -\frac{1}{n} < x < \frac{1}{n} \\ 1 & \frac{1}{n} \leq x \leq 1 \end{cases} $$
And I'm attempting to find where the sequence congerges pointwise in order to then determine uniform convergence. But I seem to be having a fundamental misunderstanding, because it seems to me that as $n \to \infty$ the ranges for my x disappear. The first domain becomes $ -1 \leq x \leq 0$, the second $0 < x < 0 $ and the third $0 \leq x \leq 1$ however that means at $ x = 0$ $f(x) = -1 \text{ or } \infty \text{ or }1$ That simply cannot be the case, unless the whole thing is simply undefined at $ x = 0$ and then my $f(x)$ becomes
$$ f(x) = \begin{cases} -1 & -1 \leq x < 0 \\ 0 & x = 0 \\ 1 & 0 < x \leq 1 \end{cases} $$
What am I misunderstanding here?