Let $X,Y$ be metric spaces ; let $\{x_n\}$ be a convergent sequence , of mutually distinct terms , in $X$ with limit $x$ and let $\{y_n\}$ be a convergent sequence in $Y$ with limit $y$ . Then is the function $f:\{x\}\cup\{x_n:n \in \mathbb N\} \to \{y\}\cup\{y_n:n\in \mathbb N\}$ defined as $f(x_n):=y_n , \forall n \in \mathbb N ; f(x):=y$ continuous ?
Continuity of function sending a convergent sequence to another convergent sequence
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metric-spaces
continuity
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0What are your thoughts on it? – 2017-01-20
1 Answers
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Yes. The only cluster point of $\{x_n: n\in \Bbb N\} \cup \{x\}$ is $x$, so we only need to study the continuity at $x$. Let $\epsilon > 0$ and choose $n_0$ such that for all $n\ge n_0$, $d(y_n, y) < \epsilon$, i.e. $d(f(x_n),f(x)) < \epsilon$. If for each $n < n_0$, $x_n \neq x$, choose $\delta>0$ such that if $d(x_n,x) < \delta$, then $n \ge n_0$, and the result follows. Otherwise, say $x=x_t$ for some $1 \le t < n_0$, then $x \neq x_n$ for all $n\neq t$. Choose $\delta > 0$ such that if $n\neq t$ and $d(x_n,x) < \delta$ then $n\ge n_0$, and we are done.