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I know that if $f$ and $g$ are both continuous then $f+g$ is continuous. I also know that there are discontinuous functions whose sum is continuous. However, I want to find two functions that are everywhere discontinuous, yet their sum is continuous. I can only come up with examples such as $f(x) = sgn x$, $g(x) =-sgn x$, but these only have one discontinuity point.

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    $\begingroup$ Take $f$ any everywhere discontinuous and $g=-f$. $\endgroup$ – Wojowu Mar 2 at 14:25
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    $\begingroup$ Try Dirchilet type functions. $\endgroup$ – Dbchatto67 Mar 2 at 14:26
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Take

$$ f(x) = \left\{ \begin{array}{ll} 1 & \quad x \in \Bbb Q \\ 0 & \quad x \in \Bbb R \setminus \Bbb Q \end{array} \right. $$

and

$$ g(x) = \left\{ \begin{array}{ll} 0 & \quad x \in \Bbb Q \\ 1 & \quad x \in \Bbb R \setminus \Bbb Q \end{array} \right. $$

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You only need to find a function $f$ that's discontinuous everywhere, since then you can take $g=-f$ and clearly $f+g=0$ will be continous.

Such a function can be found by taking $f(x)=1$ if $x$ is a rational number, and $f(x)=0$ otherwise.

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To generate infinitely many examples, take $g = h - f$, where $h$ is a continuous function.

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