If two functions f,g are both discontinuous at some point a in the domain, does this mean that the sum f+g is also discontinuous at a?

Question

Real analysis question. Having trouble either coming up with a counterexample where the sum would be continuous at a, or proving that the sum would be discontinuous at a for all cases.

Consider $f(x)=1/x, g(x)=-1/x$ and their sum at $x=0$. – user Mar 26 '21 at 16:01 — user, Mar 26 '21 at 16:01

score 3 · Answer 1 · answered Mar 26 '21 at 16:05

3

Take any fonction $f$ that is discontinuous at a point $x$. Then so is $g = -f$. But $f+g = 0$ is continuous everywhere.

answered Mar 26 '21 at 16:05

Didier

19,132

score 2 · Answer 2 · answered Mar 26 '21 at 16:02

2

Pick $$f(x)= \begin{cases}1 & \text{if } x \neq 0, \\ 0 & \text{if } x =0. \end{cases}$$

and

$$g(x)= \begin{cases}0 & \text{if } x \neq 0, \\ 1 & \text{if } x =0. \end{cases}$$

answered Mar 26 '21 at 16:02

Son Gohan

4,397

If two functions f,g are both discontinuous at some point a in the domain, does this mean that the sum f+g is also discontinuous at a?

2 Answers2