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Let $(f_n)_{n\in \mathbb N}$ be a sequence of smooth functions converging to some $f$.

Under what circumstances can I exchange limit and derivative?, i.e.

$$\lim_{n\rightarrow \infty} \frac{\partial f_n(x)}{\partial x} = \frac{\partial f(x)}{\partial x}$$

Martin Sleziak
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madison54
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  • The best result I know for this is what user57 mentions in his answer. Alas, it requires a very strong condition, namely: the sequence of derivative functions of $,{f_n},$ has to converge uniformly... – DonAntonio Jun 02 '13 at 10:28

1 Answers1

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If you have a sequence of functions $(f_n)_{n\in \mathbb N}$ that are differentiable, and converge pointwise for some point $x_o$, and if their derivatives converge uniformly, say on a given interval [a,b], supposing they are real valued functions, then the sequence of functions $(f_n)_{n\in \mathbb N}$ is uniformly convergent to $f$ and what more, $$\lim_{n\rightarrow \infty} \frac{\partial f_n(x)}{\partial x} = \frac{\partial f(x)}{\partial x}$$

This is a standard theorem in Analysis. See Walter Rudin's Principle of Mathematical Analysis, 3rd Edition, Theorem $7.17$ for detailed proof.

Dinesh
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