Question

Explain why f(x)= {x^2 if x=0, 1 if x=0 , is not continuous at x = 0. Picture provided below.

Answer 1

$f(x)$ is continuous at $x=c$ if for any $\varepsilon>0$, there exists $\delta>0$ such that

$|x-c|$

which is identical to the statement

$\displaystyle\lim_{x\to c}f(x)=f(c)$

But as $x\to0$, we have $x^2\to0$ yet $f(0)=1$, therefore $f$ is not continuous at $x=0$. The answer would be A.