Lagmental Vicfred

Uniform Derivative Convergence Recovers a Differentiable Limit by Vicfred

tags: mathematics, uniform convergence, compactness, approximation

Convergence at one point plus uniform convergence of derivatives determines a differentiable limiting function. I will separate the object being defined from the consequence being claimed.

Notation

A sequence \(f_n:X\to\mathbf R\) converges uniformly to \(f\) when \(\sup_{x\in X}|f_n(x)-f(x)|\to0\). The supremum norm captures one error bound valid everywhere.

$$ f_n(x_0)\ \text{converges},\qquad f_n'\to g\ \text{uniformly} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ f_n\to f\ \text{uniformly},\qquad f'=g $$

Stress the formula

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ f_n(x)=f_n(x_0)+\int_{x_0}^{x}f_n'(t)\,dt\longrightarrow L+\int_{x_0}^{x}g(t)\,dt $$

Interpretation

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f_n(x_0)\ \text{converges},\qquad f_n'\to g\ \text{uniformly},\\[5pt] \mathsf{C}\;&:\quad f_n\to f\ \text{uniformly},\qquad f'=g. \end{aligned} $$

Limit of the argument

Pointwise convergence allows the bad point to move with \(n\). Continuity, integration, and differentiation survive limits under different uniform hypotheses.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f_n\to f\ \text{uniformly},\qquad f'=g \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Mon 13 July 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.