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Question

(B) function $f(x)$ is continuous at $x=\alpha$ if and only if $\lim _{x \rightarrow \alpha^{-}} f(x) = \lim _{x \rightarrow \alpha^{+}} f(x) = f(\alp

Vedclass · Accepted Answer

$\lim _{x ightarrow \alpha} f(x) 
eq f(\alpha)$

Vedclass · Answer

(B) function $f(x)$ is continuous at $x=\alpha$ if and only if $\lim _{x ightarrow \alpha^{-}} f(x) = \lim _{x ightarrow \alpha^{+}} f(x) = f(\alpha)$.Since the function $f(x)$ is given as discontinuous at $x=\alpha \in(a, b)$,the condition for continuity is violated.This means that either the limit does not exist,or the limit exists but is not equal to $f(\alpha)$.Therefore,the condition $\lim _{x ightarrow \alpha} f(x) 
eq f(\alpha)$ must hold for the function to be discontinuous at that point.Option $(A)$ represents the condition for continuity,which is false here.Option $(B)$ correctly identifies the condition for discontinuity at $x=\alpha$.Options $(C)$ and $(D)$ involve limits outside the domain $[a, b]$,where the function is not defined,making them irrelevant or incorrect.

If a function $f(x)$ defined on $[a, b]$ is discontinuous at $x=\alpha \in(a, b)$,then

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