Greatest integer function [x]: a jump at every integerमहत्तम पूर्णांक फलन [x]: हर पूर्णांक पर छलाँगGreatest integer function [x]: har integer par jump

NCERT Fig 5.8 · Class 12 Maths · Chapter 5 · interactive diagram

Parts and what they mean

• x-axis (x-अक्ष) — The horizontal number line; inputs x are read here.
• y-axis (y-अक्ष) — The vertical number line; outputs y = f(x) are read here.
• Origin O(0, 0) (मूल बिंदु O(0, 0)) — Where the two axes cross.
• A flat step (एक समतल सीढ़ी) — On [k, k + 1) the value is the integer k — continuous inside each step.
• [x] = 1 for 1 \le x < 2 (1 \le x < 2 के लिए [x] = 1) — Greatest integer less than or equal to x.
• Filled dot (left end of a step) (भरा बिंदु (सीढ़ी का बायाँ सिरा)) — At an integer c, f(c) = c and RHL = c.
• f(1) = 1 (f(1) = 1) — Value and RHL at 1 are 1, but LHL at 1 is 0.
• Open dot (right end of a step) (खुला बिंदु (सीढ़ी का दायाँ सिरा)) — Just left of an integer c the value is c - 1, so LHL = c - 1 \ne c.
• \lim_{x \to 2^-} [x] = 1 (\lim_{x \to 2^-} [x] = 1) — But f(2) = 2: discontinuous at 2 (Example 15).

Original diagram drawn by Simple Learning Hub, based on NCERT Fig 5.8. Report a mistake