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CUET
-- Mathematics - Section B1
Continuity and Differentiability
The function $f(x)=[x]^2-\left[x^2\right]$ (where [y] is the greatest integer less than or equal to y), is discontinuous at |
all integers all integers except 0 and 1 all integers except 0 all integers except 1 |
all integers except 1 |
We know that the function [x] is discontinuous at all integer points. Therefore, $[x]^2$ is discontinuous at all integer points. Also, $\left[x^2\right]$ is discontinuous at $x= \pm \sqrt{|n|}$, where $n \in Z$. We observe that $\lim\limits_{x \rightarrow 1^{-}} f(x)=\lim\limits_{h \rightarrow 0} f(1-h)=\lim\limits_{h \rightarrow 0}[1-h]^2-\left[(1-h)^2\right]$ $\Rightarrow \lim\limits_{x \rightarrow 1^{-}} f(x)=0-0=0$ $\lim\limits_{x \rightarrow 1^{+}} f(x)=\lim\limits_{h \rightarrow 0} f(1+h)=\lim\limits_{h \rightarrow 0}[1+h]^2-\left[(1+h)^2\right]$ $\Rightarrow \lim\limits_{x \rightarrow 1^{+}} f(x)=1-1=0$ and, $f(1)=[1]^2-\left[1^2\right]=0$ ∴ $\lim\limits_{x \rightarrow 1^{-}} f(x)=\lim\limits_{x \rightarrow 1^{+}} f(x)=f(1)$ So, f(x) is continuous at x = 1. f(x) is discontinuous at all other integer points. |