Find all the points of discontinuity of the greatest integer func | Filo
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Class 12

Math

Calculus

Continuity and Differentiability

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Find all the points of discontinuity of the greatest integer function defined by , where [x] denotes the greatest integer less than or equal to x.

Solution: Solution: f(x)=[x]
Case 1
let there be a constant (k) which is not a integer
\displaystyle\Lim_{{{x}\to{k}}}{f{{\left({x}\right)}}}=\Lim_{{{x}\to{k}}}{\left[{x}\right]}={k}={f{{\left({k}\right)}}}
so, f(x) is continous which is not a integer
Case 2
let there be a constant (c) ehich is a integer
\displaystyle\Lim_{{{x}\to{c}^{+}}}{f{{\left({x}\right)}}}=\Lim_{{{h}\to{0}}}{f{{\left({c}+{h}\right)}}}=\Lim_{{{h}\to{0}}}{\left({c}+{h}\right)}={c}
\displaystyle\Lim_{{{x}\to{c}^{{-}}}}{f{{\left({x}\right)}}}=\Lim_{{{h}\to{0}}}{f{{\left({c}-{h}\right)}}}=\Lim{\left({c}-{h}\right)}={c}-{1}
Left Hand LimitRight Hand Limit
so, all integers are not continous.
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