How to preserve continuity in the limit of a sequence of continuous functions.

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Concept |
Content |

Sequences of
Bounded Functions |
The concept of sequences but for functions instead of real numbers. |

Pointwise Convergence |
A notion of convergence for sequences functions that reduces the question of convergence to convergence of sequences of real numbers. |

Uniform Convergence |
A strong notion of convergence for sequences of functions that helps to preserve favorable properties like continuity in the limit. |

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

Epsilon-Delta
Definition |
A different notion of continuity using open intervals. If the input to a continuous function varies less than delta, then the output values should vary less than epsilon. |

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Now we show that a uniformly convergent sequence of continuous
functions has to converge to a continuous function.

*Proof:* Let \(\varepsilon>0\) and let \(x_0\in I\). Since \((f_n)_{n\in\mathbb{N}}\) converges
uniformly to \(f:I\to\mathbb{K}\),
there exists some \(N\) such that for
all \(n\geq N\) and \(x\in I\) holds \[|f(x)-f_n(x)|<\frac\varepsilon3.\]
Since \(f_n\) is continuous on \(I\), there exists some \(\delta>0\) such that for all \(x\in I\) with \(|x-x_0|<\delta\) holds \[|f_n(x_0)-f_n(x)|<\frac\varepsilon3.\]
Altogether, we then have \[\begin{aligned}
|f(x)-f(x_0)|=&\,|f(x)-f_n(x)+f_n(x)-f_n(x_0)+f_n(x_0)-f(x_0)|\\
\leq &\,|f(x)-f_n(x)|+|f_n(x)-f_n(x_0)|+|f_n(x_0)-f(x_0)|\\
<
&\,\frac\varepsilon3+\frac\varepsilon3+\frac\varepsilon3=\varepsilon.
\end{aligned}\] Therefore, \(f\)
is continuous by the \(\varepsilon\)-\(\delta\) criterion. \(\Box\)

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