Suppose that $\left\{f_n\right\}$ converges uniformly to $f$, and that each $f_n$ has at most $M$ discontinuities, where $M\in \mathbb{N}$ is a fixed value. The $f_n$ don't need to be discontinuous at the same points.
QUESTION: Does it necessarily follow that $f$ has at most $M$ discontinuities?
This occurred to me while I was taking a walk, and I was wondering if the above statement was true or if there is a counterexample(possibly pathological).
Best Answer
Yes $f$ has at most $M$ discontinuities. If $f$ has a discontinuity at $p$, there must be $\epsilon > 0$ such that in any neighbourhood of $p$ there are points $x, y$ with $|f(x) - f(y)| > \epsilon$. If $f_n \to f$ uniformly, for sufficiently large $n$ we have $|f_n - f| < \epsilon/3$, and then the condition of the last sentence holds for $f_n$ with $\epsilon$ replaced by $\epsilon/3$, so $f_n$ is also discontinuous at $p$.