Exercise 2.2.4

Question

Let $\pi_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}$ and $\pi_{2}: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be the functions $\pi_{1}(x, y):=x$ and $\pi_{2}(x, y):=y$. Show that $\pi_{1}$ and $\pi_{2}$ are continuous. Conclude that if $f: \mathbb{R} \rightarrow X$ is any continuous function into a metric space $(X, d)$, then the functions $g_{1}: \mathbb{R}^{2} \rightarrow X$ and $g_{2}: \mathbb{R}^{2} \rightarrow X$ defined by $g_{1}(x, y):=f(x)$ and $g_{2}(x, y):=f(y)$ are also continuous.

Prakash Anand · Accepted Answer

First note that the second part generalizes the first (just take $f : ℝ \to ℝ$ defined by $f (x) = x)$ . So we will prove only the second part (the proofs are basically the same anyway).

Proof. Let $y = (x_{1}, y_{1}) \in ℝ^{2}$ and let $𝜖 > 0$ be given. Since $f$ is continuous, we know that there is some $δ > 0$ such that if $d_{ℝ} (x, x_{0}) = | x - x_{0} | < δ$ then $d_{X} (f (x), f (x_{0})) < 𝜖$ .

If $d_{(ℝ^{2}, l^{2})} ((x_{1}, y_{1}), (x_{2}, y_{2})) < δ$ (note that this is the same $δ$ as above), then

\begin{aligned} \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}} < δ \\ \Rightarrow {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} < δ^{2} \\ \Rightarrow {(x_{2} - x_{1})}^{2} < δ^{2} and {(y_{2} - y_{1})}^{2} < δ^{2} \\ \Rightarrow | x_{2} - x_{1} | < δ and | y_{2} - y_{1} | < δ \\ \Rightarrow d_{X} (f (x_{1}), f (x_{2})) < 𝜖 and d_{X} (f (y_{1}), f (y_{2})) < 𝜖 (from continuity of f) \\ \Leftrightarrow d_{X} (g_{1} (x_{1}, y_{1}), g_{1} (x_{2}, y_{2})) < 𝜖 and d_{X} (g_{2} (x_{1}, y_{1}), g_{2} (x_{2}, y_{2})) < 𝜖 (by definition) \end{aligned}

Thus $g_{1}, g_{2}$ are continuous. □

Note that we can also deduce the second part from the first with the following argument: We have $f : ℝ \to X, π_{i} : ℝ^{2} \to ℝ$ both continuous, so take the composition $g : ℝ^{2} \to X$ defined by $g_{i} (x, y) = f \circ π_{i} (x, y)$ , and we know that the composition of continuous functions is continuous, and this is what we wanted.

mathchakra

2021-12-19 18:14

Exercise 2.2.4

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Exercise 2.2.4

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