MathDB

Problem 3

Part of 1995 VJIMC

Problems(2)

f>f^(-1) if f and f^(-1) are decreasing

Source: VJIMC 1995 1.3

11/15/2021
Let f(x)f(x) and g(x)g(x) be mutually inverse decreasing functions on the interval (0,∞)(0,\infty). Can it hold that f(x)>g(x)f(x)>g(x) for all x∈(0,∞)x\in(0,\infty)?
function
f(x)=g(x)sin+h(x)cos with continuous functions

Source: VJIMC 1995 2.3

11/25/2021
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?
functioncontinuousreal analysis