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Topology: An Inquiry-Based Approach

Steven Schlicker

Section Composites of Continuous Functions

Let \((X, d_X)\text{,}\) \((Y,d_Y)\text{,}\) and \((Z, d_Z)\) be metric spaces, and suppose \(f: X \to Y\) and \(g: Y \to Z\) are continuous functions. It seems natural to ask if the composite \(g \circ f : X \to Z\) is a continuous function.

Activity 6.4.

Let \((X, d_X)\text{,}\) \((Y,d_Y)\text{,}\) and \((Z, d_Z)\) be metric spaces, and suppose \(f: X \to Y\) and \(g: Y \to Z\) are continuous functions. We will prove that \(g \circ f\) is a continuous function.

(a)

What do we have to do to show that \(g \circ f\) is a continuous function? What are the first two steps in our proof?

(b)

Let \(a \in X\) and let \(b = f(a)\text{.}\) Suppose \(\epsilon \gt 0\) is given. Explain why there must exist a \(\delta_1 \gt 0\) so that \(d_Y(y,b) \lt \delta_1\) implies \(d_Z(g(y), g(b)) \lt \epsilon\text{.}\)

(c)

Now explain why there exists a \(\delta_2 \gt 0\) so that \(d_X(x,a) \lt \delta_2\) implies that \(d_Y(f(x), f(a)) \lt \delta_1\text{.}\)

(d)

Prove that \(g \circ f : X \to Z\) is a continuous function.
Continuity is an important concept in topology. We have seen how to define continuity in metric spaces, and we will soon expand on this idea to define continuity without reference to metrics at all. This will allow us to later define continuous functions between arbitrary topological spaces.
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