How combination of continuous functions leads to new continuous functions.

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Concept |
Content |

Composition |
The composition for maps is just applying two maps in a row. |

Limit Theorems |
Combining limits is a useful tool to deduce convergence and the limit of a more complicated sequence from the convergence of simpler building blocks. |

Continuity |
The concept that relates functions with convergent sequences. |

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Next we give a result on the continuity of sums, products and
quotients of functions. This looks very similar to the limit theorem for
sequences. Indeed, those results on sums, products and quotients of
sequences are the “main ingredients” for the proof (which is therefore
skipped).

Let \(I\subset \mathbb{K}\) and let \(f,g:I\to\mathbb{K}\) be continuous in \(x_0\in I\). Then also \(f+ g\) and \(f\cdot g\) are continuous in \(x_0\). Furthermore if \(g(x_0)\neq0\), then also \(\frac{f}g\) is continuous in \(x_0\).

Now we consider the *composition of functions \(f\) and \(g\)* (\(f\circ g\)) which is defined by the formula
\((f\circ g)(x)=f(g(x))\).

Let \(I_1,I_2\subset\mathbb{K}\) and
\(f:I_1\to\mathbb{K}\), \(g:I_2\to\mathbb{K}\) be functions with
\[g(I_2)=\{g(x)\,:\,x\in I_2\}\subset
I_1.\] Assume that \(g\) is
continuous in \(x_0\in I_2\) and \(f\) is continuous in \(g(x_0)\in I_1\). Then also \(f\circ g\) is continuous in \(x_0\).

*Proof:* Let \((x_n)_{n\in\mathbb{N}}\) be a sequence in
\(I_2\) with \(\lim_{n\to\infty}x_n=x_0\). By the
continuity of \(g\) holds \(\lim_{n\to\infty}g(x_n)=g(x_0)\). Then, by
the continuity of \(f\) holds \[\lim_{n\to\infty}f(g(x_n))=f(g(x_0)).\]
\(\Box\)

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