What is uniform convergence for one function and how is that equivalent to continuity? My teacher wanted to prove that function
$$\frac{1}{x} - \sum_{n=1}^\infty \frac{2x}{n^2-x^2}$$
is continuous. He said that it is equivalent to proving that
$$\sum_{n=1}^\infty \frac{1}{n^2-x^2}$$
uniformly converges. However uniform convergence is defined for a sequence of function as I understand it. Here I see only one function: the equation depends purely on x and nothing else than x - that is just one function in terms of x.
But even if it made sense, how is uniform convergence equivalent to continuity?
According to my knowledge he didn't even prove uniform convergence (despite saying that). He only proved that the equation above:
$$\sum_{n=1}^\infty \frac{1}{n^2-x^2}$$
converges (to a number smaller than something). And convergence (as I understand it) is different from uniform convergence. 
How is that correct?

Theorem: the uniform limit of a sequence of continuous functions on an interval is a continuous function on that interval.  
If you examine the teacher's proof that $\sum_{n=1}^\infty \frac{1}{n^2 - x^2}$ converges (presumably on some interval that doesn't contain any positive integer $n$), it may have in it some estimate that is true for all 
$x$ in the interval.  That will allow you to conclude that the convergence is
uniform.
