Let h(x)=|x|

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

Problem

Let

h(x)=|x|\,\operatorname{sgn}(x-2)-\lfloor x-2.5\rfloor.

Answer the questions that follow. (i) Evaluate

h(x)

at the points

x=1.9,\;2,\;2.5,\;3.2

and

x=-4

. (ii) Using your results from part (i), find the number of integers

k

in the interval

[-4,4]

for which

h(k)=0

. Give proper reasoning. (iii) Show that

h(x)

is discontinuous at

x=2

and at

x=2.5

by computing the left-hand and right-hand limits at each of these points.