Continuity of the limit of a function I am having a hard time getting a foothold on this question. I feel like discontinuity must occur at 0, but I cannot get to a proof. The question is below: Suppose that $f_n$ is the function from $[0,1]$ to $[0,1]$ whose graph is a straight line joining the points $(0,1)$ and $(\frac{1}{n}, 0)$ together with the straight line joining $(\frac{1}{n},0)$ and $(1,0)$. Then each $f_n$ is a continuous function, as you can see from its graph. For each $x \in [0,1]$, the limit $\lim_{n \to \infty} f_n(x)$ exists and we define $g(x) = \lim_{n \to \infty} f_n(x)$. Show that $g(x)$ is not continuous.

What is $g(0)$? What is $g(x)$ for any other $x$? Do we have $\lim_{x\to 0} g(x) = g(0)$?