This is a subjective problem from TOMATO based on Graphing integer value function.
Problem: Graphing integer value function
Let [x] denote the largest integer (positive, negative or zero) less than or equal to x. Let be defined for all real numbers x.
(i) Sketch on plain paper, the graph of the function f(x) in the range
(ii) Show that, any given real number , there is a real number such that
First note that is same as .
It’s graph between 0 to 1 looks like:
Clearly [x] part only increments (or decrements) it by integer quantity as [x] is constant between any two integers. That for any integer k for all ( x \in (k, k+1) ).
, . Hence graph of f(x) is as follows:
Finally consider and arbitrary value . We take . Then (since )