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hey guys I was wondering if anyone could offer me a few hints for a question on continuity.($f,g$ both $\mathbb R$ to $\mathbb R$ functions, and considers the point $a$ which is an integer)

If neither $f$ nor $g$ is continuous at $a$, then $f+g$ is not continuous at $a$. (true or false question quick explanation required)

So for this question I understand how to do if it were the other way round when $f$ and $g$ are continuous at $a$, and $f+g$ is continuous at $a$.

something like this (http://prntscr.com/ho31yn).

However I'm not really sure how to do the other one.

All help appreciated.

BDN
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Mathlete
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2 Answers2

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Take any $f$ discontinuous and $g:=-f$, also discontinuous.

In fact $f$ and $g$ must verify $f+g=h$ where $h$ is continuous. Hence all pairs $f,h-f$ will do !

  • A short answer would probably eschew announcing itself as such. – user121330 Dec 15 '17 at 22:32
  • @user121330: I know but there is a serious risk to be closed as "not sufficient for an answer". On second thoughts I make it longer. –  Dec 15 '17 at 22:34
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It seems intuitively true, but the statement is wrong. Consider $$ f(x)=\begin{cases}1 & x\geq 0\\0 & x<0\end{cases} \text{ and } g(x)=\begin{cases}0 & x\geq 0\\1 & x<0\end{cases}. $$ Then $f$ and $g$ are discontinuous at $0$ but $f+g\equiv 1$ is continuous at $0$.

(And also $fg\equiv 0$ is continuous at $0$)

Mundron Schmidt
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