Sequence of continuous function that converges pointwise

Question

Specify the sequence of continuous function $f_{n} :\mathbb R \to \mathbb R, n\in \mathbb N$ that converges pointwise to not continuous function $f :\mathbb R \to \mathbb R$.

Answer 1

One example:

$$f_n(x) = \begin{cases} 0 & \text{if } x \leq 0\\ nx & \text{if } 0<x<\frac{1}{n}\\ 1 & \text{if } x \geq \frac{1}{n} \end{cases} $$

Then $f_n \to f = \chi_{(0, \infty)}$ pointwise, because for any $x\in\mathbb{R}$, there exists an $N$ such that $|f_n(x) - f(x)| = 0$ for all $n \geq N$ (draw a graph if you are not sure).

In fact, you can draw a lot of similar sequences of functions. Please write your own solution and argument in your homework.

Answer 2

Define the functions $f_n$ by

$$\forall |x|>\frac1n, f_n(x) = 0$$ $$\forall |x| < \frac1n, f_n(x) = (1 - nx)(1 + nx)$$

Then $f_n$ is continuous, but $f = \lim_{n\to\infty} f_n$ is not, as $f(x) = 0$ for all $x \ne 0$, and $f(0) = 1$.