MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions
Problem
Consider the functions u(x)=⌊x⌋ and v(x)=sgn(x).
(i) Using their definitions, explain why each function possesses jump discontinuities.
(ii) Identify all points of discontinuity of u(x) and of v(x). Support your answer by evaluating the left-hand and right-hand limits at those points.
(iii) Compare the magnitude of the jumps at the discontinuities of u(x) with that at the discontinuity of v(x).