Question

If $f(x)=x^2$, then

• A. $f$ is continuous at $x=0$
• B. $f$ is discontinuous at $x=0$
• C. $f$ is not defined at $x=0$
• D. None of these

Answer 1

Answer: (A)

First note that the function is defined at the given point $x=0$ and its value is 0 .
Then, find the limit of the function at $x=0$. Clearly,
$\displaystyle\lim _{x \rightarrow 0} f(x)=\displaystyle\lim _{x \rightarrow 0} x^2=0^2=0$
Thus, $\displaystyle \lim _{x \rightarrow 0} f(x)=0=f(0)$
Hence, $f$ is continuous at $x=0$.