Sequence of functions

Give an example of a sequence of functions $f_n:\mathbb{R} \rightarrow \mathbb{R}$ such that each $f_n$

1.  is continuous in $(-\infty, 0)$
2.  is discontinuous in $(0, +\infty)$
3. converges uniformly to a continuous function

Solution



We take:

$$f_n(x)=\left\{\begin{matrix}
0 &,  &x<0 \\
 0&,  & x \in \mathbb{Q}\cap [0, +\infty) \\
 \frac{1}{n}& , & x \in \mathbb{R}\setminus \mathbb{Q}\cap (0, +\infty)
\end{matrix}\right.$$

and we see that it converges uniformly to the zero function, since it has a distance of $1/n$.