Compatibility of differentiability with uniform limits.

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Concept |
Content |

Power Series |
A sequence of partial sums of polynomial functions. |

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

Differentiability |
How to quantify the rate of change of a function. |

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. |

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Let \(I \subseteq \mathbb{R}\) and
\((f_1, f_2, f_3, f_4, f_5,\dots)\) a
sequence of functions \(f_n \colon I \to
\mathbb{R}\). Recall that, if the functions \(f_n\) are continuous and there exists a
further function \(f\colon I \to
\mathbb{R}\) such that \(f_n\)
converges *uniformly* to \(f\),
then \(f\) is also continuous.

Here, we want to deal with the question, if a uniformly converging
sequence of *differentiable* functions, also has a differentiable
limit function \(f\).

Let \((f_1, f_2, f_3, \dots)\) be a
sequence of functions \(f_n \colon I \to
\mathbb{R}\). Assume that

\((f_n)_{n \in \mathbb{N}}\) is
pointwisely convergent to a function \(f
\colon I \to \mathbb{R}\).

\(f_n \colon I \to \mathbb{R}\)
is differentiable for all \(n \in
\mathbb{N}\).

There exists \(g \colon I \to
\mathbb{R}\) with \(\|f_n^\prime - g
\|_\infty \to 0\) for \(n \to
\infty\), i.e. the sequence of derivatives \((f_n^\prime)\) converges uniformly to \(g\).

Then \(\|f_n - f\|_\infty \to 0\)
for \(n \to \infty\) and \(f\) is differentiable with \(f^\prime = g\).

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