Measure, Integration & Real Analysis · Problem 6A.12

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · metric-spaces-composition-continuous

Problem

Suppose

(U, d_U)

(V, d_V)

, and

(W, d_W)

are metric spaces. Suppose also that

T: U \to V

and

S: V \to W

are continuous functions. (a) Using the definition of continuity, show that

S \circ T: U \to W

is continuous. (b) Using the equivalence of 6.11(a) and 6.11(b), show that

S \circ T: U \to W

is continuous. (c) Using the equivalence of 6.11(a) and 6.11(c), show that

S \circ T: U \to W

is continuous.