How would these functions differ given the following property?

Question

Function a: $|x−1|<0.5=>|a(x)−3|<2$ and that the limit of this function as x approaches 1 exists.

Function b: $0<|x−1|<0.5=>|b(x)−3|<2$ and that the limit of this function as x approaches 1 exists.

Does the $0<$ in function b make the function any different from function a? If so, how?

Answer 1

The only difference is that $b(x)$ does not need to be defined at $x=1$ or if it is it can be anything. As an example, let $c(x)=x+2$. This clearly satisfies the requirement for $a$. Let $$d(x)=\begin {cases} x+2& x \neq 1\\1000&x=1 \end {cases}$$ It satisfies the requirement for $b$, but fails the requirement for $a$.