Question

A function $f\left(x\right)$ is continuous at a point $x=a$, then which of the following is incorrect.

• A. $f(a+0.0001)=f(a-0.0001)$
• B. $f\left(a\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$
• C. $\underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)$
• D. $\underset{h\to 0}{lim}f(a-h)=f\left(a\right)$

Answer 1

The correct option is A

$f(a+0.0001)=f(a-0.0001)$

Option b. ,c. ,d. are correct. This deals with the condition by which we say a function is continuous.

The condition are
1. A finite limit exist for the function at the given point, $\underset{x\to a}{lim}f\left(x\right)$

2. $\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$

These conditions are represented by option b. ,c. ,d.
$f(a+0.0001)=f(a-0.0001)$ doesn't mean that limit exist and for a continuous function this may not be the case. The $\Delta$ in $f(a+\Delta )$ or $f(a-\Delta )$ should be infinitesimally small, but 0.0001 is very large compared to such a desired value.

Hence (a) is incorrect.