Uniform Convergence for Differentiable Functions
Compatibility of differentiability with uniform limits.
Discover Bridges
Study Uniform Convergence for Differentiable Functions
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 \(f^\prime = g\).
Discuss your questions by typing below.
Solve the Exercise
The data for the interactive network on this webpage was generated with pntfx Copyright Fabian Gabel and Julian Großmann. pntfx is licensed under the MIT license. Visualization of the network uses the open-source graph theory library Cytoscape.js licensed under the MIT license.