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Let $A\subseteq \mathbb{R}$, function $f\colon A\to \mathbb{R}$ continuous at $a\in A$, and $g\colon A \to \mathbb{R}$ is discontinuous at $a\in A$.
Prove or disprove that $f+g$ is discontinuous at $a\in A$.

A proof by definition is what I need. But if ones try with the other way that's okay. Help me.

Thank you.

Sal
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plifendm
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    Welcome to MSE. what did you try? –  Dec 14 '14 at 08:12
  • Do you know the theorem that the sum of two continuous functions is continuous? – Potato Dec 14 '14 at 08:16
  • @Potato Yes, I do. But my bad when I was on my exam I didn't even notice that theorem. – plifendm Dec 15 '14 at 09:18
  • @DanisFischer I wrote out the def of continuous f and discontunious g. I supposed that f+g continuous then found a contradiction. My lecturer gave me score 6 of 10 for this problem. I know there were some "prompted" steps in my answer. lol. – plifendm Dec 15 '14 at 09:24

2 Answers2

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You may already know that the sum and the difference of two continuous functions are continuos. Let $f$ be continuous, $g$ not continuous and let $h=f+g$. If we assume tha t$h$ is continuous, then by what I just mentioned, $g= h-f$ is also continuous, contradiction. Hence $h$ cannot be continuous.

Hagen von Eitzen
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Let $f=x, g = \frac{1}{x}$, then $f$ continous at $0$, but $g$ isn't. And $f+g = x+\frac{1}{x}$ which is still not continous at $0$.

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Mr.Fry
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