A digital thermometer shows only the integer part of the actual body temperature in…

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

A digital thermometer shows only the integer part of the actual body temperature in degrees Celsius. If the actual temperature is

x

, the displayed temperature is

T(x)=\lfloor x\rfloor

. A warning indicator lights up according to

W(x)=\operatorname{sgn}(x-38)

. Define

F(x)=T(x)+W(x)

for

x\in\mathbb{R}

. (i) Obtain explicit rules for

F(x)

on the intervals

37\le x<38

and

38\le x<39

. (ii) Calculate

F(37.4)

F(38)

and

F(38.9)

. (iii) Find all real numbers

x

such that

F(x)=40

. Express your answer in interval form.