SOLUTION: Given the integer function f(x)= int(2x), find A to D. A. Find domain. B. Find range. C. Locate intercepts. D. If f continuous on its domain?

It is well known fact that the parent function int(x) is piecewise constant.

Making leaps at every integer point on x-axis, it takes a constant value of integer number n 
in each interval  n <= x < n+1  and is constant inside every such interval.



From it, it is clear that function f(x) = int(2x) is also piecewise constant.

Making leaps, it takes a constant value of integer number n in each half-integer interval 
over x-axis and is constant inside every such interval.


Same as the function int(x) is defined over the entire number line, function int(2x) 
is defined over the entire number line.  So, its domain is the set of all real numbers.

        Thus, question (A) is answered.



Its range is the set of all integer numbers (a discrete set).

        Thus, question (B) is answered, too.



Regarding intercepts.

y-intercept is  y = int(2*0)= int(0) = 0.  In other words, y-intercept is the point y= 0 at vertical y-axis x= 0,
             or the point (0,0), the origin of the coordinate system.


x-intercept is the set of solution to equation y= 0.  

It is the set of points (x,0) on x-axis y= 0,  where  0 <= x < 1/2.

In other words, x-intercept is horizontal interval [0,1/2) on horizontal x-axis.

        So, question (C) is answered, too.



The answer to question (D) is obvious.

From what I said at the beginning, function f(x) = int(2x) is piecewise constant over every one half-a-unit interval
on x-axis.  Such a function is INEVITABLY non-continuous. It fails/loses continuity at every half-integer point over x-axis.