Let f(x) = z - |p|, where z >= 0 and |p| is the greatest integer not larger than z. Then f(x) is a: A. monotonically increasing function B. monotonically decreasing function C. linearly increasing function between two integers D. linearly decreasing function between two integers

GATE 2012 · General Aptitude · Functions · medium

Answer: f(z) = z - floor(z) is linearly increasing between consecutive integers (fractional part / sawtooth function).

1. Simplify on an integer interval: f(z) = z - n is a linear function with slope 1 (linearly increasing) on the interval [n, n+1). The function rises from f(n) = 0 up to f approaches 1 as z approaches n+1 from below.
2. Check other options: f is NOT monotonically increasing overall (it drops to 0 at each integer), NOT decreasing. Between integers it is linear with positive slope, so C is correct: linearly increasing function between two integers.