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f(x)=g(x)sin+h(x)cos with continuous functions

Source: VJIMC 1995 2.3

November 25, 2021
functioncontinuousreal analysis

Problem Statement

Let f:R→Rf:\mathbb R\to\mathbb R be a continuous function. Do there exist continuous functions g:R→Rg:\mathbb R\to\mathbb R and h:R→Rh:\mathbb R\to\mathbb R such that f(x)=g(x)sin⁡x+h(x)cos⁡xf(x)=g(x)\sin x+h(x)\cos x holds for every x∈Rx\in\mathbb R?
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