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

The answer is A.

Answer 2

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

`|x-c|<\delta\implies|f(x)-f(c)|<\varepsilon`

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.