Term-by-term_Differentiability_Theorem

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Definition

Let $(f_n)$ be a sequence of differentiable functions on the interval $[a,b]$, and assume $(f_n')$ converges uniformly on $[a,b]$. If there exists a point $x_0 \in [a,b]$ for which $(f_n(x_0))$ converges, then:

  1. $(f_n)$ converges uniformly on $[a,b]$
  2. The limit function $f = \lim f_n$ is differentiable
  3. $f' = \lim f_n'$ on $[a,b]$

Proof Sketch

The key is to use the Mean_Value_Theorem to show that uniform convergence of derivatives, combined with pointwise convergence at one point, implies uniform convergence of the functions.

For any $x, y \in [a,b]$:

$$ |f_n(x) - f_m(x)| \leq |f_n(x) - f_n(x_0)| + |f_n(x_0) - f_m(x_0)| + |f_m(x_0) - f_m(x)| $$

Apply MVT to bound the first and third terms using the uniform convergence of $(f_n')$.