Exercise 5.4.3

Question

Taking the continuity of $h(x)$ as given, reference the proper theorems from Chapter 4 that imply that the finite sum
$$
g_{m}(x)=\sum_{n=0}^{m} \frac{1}{2^{n}} h\left(2^{n} x\right)
$$
is continuous on $\mathbf{R}$.

Ulisse Mini · Accepted Answer

The Composition of Continuous Functions (Theorem 4.3.9) ensure $h(2^n x)$ is continuous, and the various parts of the Algebraic Continuity Theorem (Theorem 4.3.2) ensure their combinations to form $g_m(x)$ is continuous.

Exercise 5.4.3

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Exercise 5.4.3

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