Exercise 2.4.2

Question

Let $f: X \rightarrow Y$ be a function from a connected metric space $(X, d)$ to a metric space $\left(Y, d_{d i s c}\right)$ with the discrete metric. Show that $f$ is continuous if and only if it is constant.

Prakash Anand · Accepted Answer

\begin{proof}
\begin{description}
\item[$(\Rightarrow)$] Suppose that $f$ is continuous, and also suppose that $f$ is not constant. That is, there are at least two values in $Y$, say $x, y \in Y$. From the previous exercise we know that $\left(Y, d_{	ext {disc }}ight)$ is disconnected. However, this contradicts Theorem 2.4.6. Thus $f$ must be constant.

\item[$(\Leftarrow)$] Clearly if $f$ is constant it is also continuous.
\end{description}
\end{proof}

Exercise 2.4.2

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Exercise 2.4.2

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