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Let $I\subset\mathbb{R}$ be an interval and let $(f_n)$ be a sequence of continuous real-valued functions on $I$. Consider the following statements:

  1. $f_n\to f$ uniformly;
  2. For every sequence $(x_n)$ in $I$ converging to $x\in I$, we have $f_n(x_n)\to f(x)$;
  3. $f:\ I\longrightarrow\mathbb{R}$ is continuous.

Certainly (1) implies (2) and (3). If $I$ is compact, (2) and (3) together imply (1), otherwise counterexamples can be found.

My interest is on the relationship between (2) and (3), specifically does (2) imply (3)? My intuition says no, since (2) would appear to be a strictly weaker statement than (1). However, (2) is certainly stronger than pointwise convergence of $(f_n)$, and so far I have been unable to think of a counterexample. Can anyone think of a counterexample (or proof)

Jason
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1 Answers1

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The answer is positive. Indeed $2$ implies $3.$

The condition $2$ is called a continuous convergence.