# Combination Theorem for Continuous Functions/Sum Rule

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## Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $f$ and $g$ be functions which are continuous on an open subset $S \subseteq X$.

Then:

- $f + g$ is continuous on $S$.

## Proof

By definition of continuous, we have that

- $\forall c \in S: \displaystyle \lim_{x \mathop \to c} \map f x = \map f c$
- $\forall c \in S: \displaystyle \lim_{x \mathop \to c} \map g x = \map g c$

Let $f$ and $g$ tend to the following limits:

- $\displaystyle \lim_{x \mathop \to c} \map f x = l$
- $\displaystyle \lim_{x \mathop \to c} \map g x = m$

From the Sum Rule for Limits of Functions, we have that:

- $\displaystyle \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$

So, by definition of continuous again, we have that $f + g$ is continuous on $S$.

$\blacksquare$

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**continuous function**(i)