Question

Write all
points of discontinuity of
greatest integer function [x] in the interval
[-3,1).​

Answer 1

${\large{\star }} \: \: \: \: \large \bf{ \color{brown} \underline{ \underline{Explanation }} } \: \: \: \: \large \star\\ \\ \circ \: \: \tt \green{ process \: 1} \\ \\ \tt \: f(x) = [x] \: \\ \\ \tt \sharp \: \: \: to \: \: check \: \: the \: \: pts \: \: of \: \: discontinuity \: \: in \: \: its \: domain \\ \tt lets \: \: have \: \: a \: \: look \: \: in \: \: its \: \: graph \\ \\ \checkmark \tt \: from \: \: this \: \: graph \: \: we \: \: can \: say \: \: function \: \: is \: \: discontinuous \\ \tt \: \: at \: \: any \: \: natural \: \: value \: \: of \: \: x \: \: i n \: \: its \: \: domain \\ \tt \: hecne \: \: pts \: \: are \: \: 0, \: - 1, - 2, - 3 \\ \\ \tt \circ \: \: \: \green{process \: 2} \\ \\ \tt \: \natural \: \: checking \: \: pts \: \: individually \: \: with \: \: algebraic \: \: Process \\ \\ \tt \checkmark \: checking \: \: for \: \: pt \: 0 \\ \\ \tt \lim_{ x \to0^{ + } }f(x) = 0 \\ \tt \lim_{ x \to0^{ - } }f(x) = - 1 \\ \\ \therefore \: \: \tt \: \lim_{ x \to0^{ + } }f(x) \not = \lim_{ x \to0^{ - } }f(x) \\ \\ \\ \\ \tt \lim_{ x \to - 1^{ + } }f(x) = - 1\\ \tt \lim_{ x \to - 1^{ - } }f(x) = -2 \\ \\ \therefore \: \: \tt \: \lim_{ x \to - 1^{ + } }f(x) \not = \lim_{ x \to - 1^{ - } }f(x) \\ \\ \tt \checkmark similarly \: \: we \: \: can \: \: say \: \: this \: \: for \: \: other \: \: natural \: \: values\: \: of \: \: x \: \\ \tt \: hence \: pts \: \: are \: 0 , - 1, - 2, - 3$