# Combination Theorem for Continuous Functions/Real/Sum Rule

< Combination Theorem for Continuous Functions | Real(Redirected from Sum Rule for Continuous Real Functions)

Jump to navigation
Jump to search
## Theorem

Let $\R$ denote the real numbers.

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

Then:

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

## Proof

By definition of continuous:

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

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

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

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

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

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

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 17$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**continuous function**(i)