Question

Function f(x) is defined in [a, b]. If the function is continuous throughout the interval [a, b] then which among the following are correct

• A. $\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)$
• B. $\underset{x\to a}{lim}f\left(x\right)=\underset{x\to b}{lim}f\left(x\right)$
• C. $\underset{x\to b}{lim}f\left(x\right)=f\left(a\right)$
• D. $\underset{x\to a^{+}}{lim}f\left(x\right)=\underset{x\to a^{-}}{lim}f\left(x\right)$

Answer 1

The correct option is A

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

The given function is defined for [a,b]. For such a closed interval we test the continuity at the boundary

points as below. For f(x) to be continuous at x=a

$f\left(a\right)={lim}_{x\to a^{+}}f\left(x\right)$

for f(x) to be continuous at x=b

$f\left(b\right)={lim}_{x\to b^{-}}f\left(x\right)$