Question

In which of the following cases is the function f(x) discontinuous at a?

• A. $\underset{x\to a}{Lt}f\left(x\right)=\infty$
• B. $\underset{x\to a^{-}}{Lt}f\left(x\right)\ne \underset{x\to a^{+}}{Lt}f\left(x\right)$
• C. $f\left(a\right)\ne \underset{x\to a^{+}}{Lt}f\left(x\right)$
• D. $\underset{x\to a}{Lt}f\left(x\right)=0$

Answer 1

Answer: (A), (B), (C)

$f\left(a\right)\ne \underset{x\to a^{+}}{Lt}f\left(x\right)$

The condition for limit to exist a point a is,

$\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{x\to a^{-}}{lim}f\left(x\right)=L=finite$

i.e., Both LHL and RHL should be equal to each other and finite. Option (c) also gives a case when the function is discontinuous but the reason is not that limit doesn't exist but because the limiting value is not equal to the function's value at that point. In option (a) the limit value is infinite. In option (b) the LHL is given to be unequal to the RHL which by definition is a case where limit of the function doesn't exist at the point concerned.