Suppose that (xn) is a convergent sequence of points which...

Suppose that $\left(x_n\right)$ is a convergent sequence of points which lies, together with its limit $x$, in a set $D \subseteq \mathbf{R}^n$. Suppose that $\left(f_n\right)$ converges on $D$ to the function $f$. Is it true that $f(x)=\lim \left(f_n\left(x_n\right)\right)$ ?

Key Concepts

Uniform Convergence

Uniform convergence strengthens pointwise convergence by requiring that the convergence of {f?} to f occurs at the same rate for all x in D. In other words, for every ? > 0, there exists an N such that for all n ? N and for all x ? D, |f?(x) ? f(x)| < ?. This property is critical when exchanging limits because it guarantees that the limit function f inherits much of the 'nice' behavior of the f?, such as continuity.

Sequential Continuity and Limit Exchange

The principle of sequential continuity states that if a function is continuous at a point x, then for any sequence {x?} converging to x, the sequence {f(x?)} converges to f(x). When dealing with a sequence of functions that converge (especially uniformly) to f, if f is continuous at x, then one can often exchange the order of the limits (f(x) = lim??? f?(x?)). However, without uniform convergence or the continuity of f at x, this interchange may not hold.

Convergence of Sequences

This refers to the idea that a sequence {x?} in a metric space (such as ??) converges to a limit x if, for every ? > 0, there exists an index N such that for all n ? N, the distance between x? and x is less than ?. This concept is foundational in analysis, setting the stage for understanding limits and continuity.

Pointwise Convergence of Functions

A sequence of functions {f?} converges pointwise to a function f on a domain D if, for every fixed x in D, the sequence {f?(x)} converges to f(x) as n ? ?. This type of convergence, however, does not guarantee that the rate of convergence is uniform across D, which can create issues when interchanging limits.

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