so I am preparing for my final and I needed some help verifying my definitions and their justifications. So any help/feedback would be appreciated...
Let $f:(0,1) \to \mathbb{R}$ be a given function, and f is continuous. Then out of the following defnition pick the right definition, and/or equuivalent definition. If the definition is not equivalent provide explanation:
a.) for any $\epsilon \gt 0$, there exists $\delta \gt 0$ such that for all $x \in (0,1)$ and $0 \lt |x-x_0| \lt \delta$, one has $|f(x) - f(x_0)|\lt \epsilon$.
this definition is true.
b.) for any $\epsilon \gt 0$, there exists $\delta \gt 0$ such that for all $x \in (0,1)$ and $|x-x_0| \leq \delta$, one has $|f(x) - f(x_0)|\leq \epsilon$.
this definition is equivalent.
c.) for any $\epsilon \gt 0$, for any $\delta \gt 0$ such that for all $x \in (0,1)$ and $ |x-x_0| \lt \delta$, one has $|f(x) - f(x_0)|\lt \epsilon$.
this definition is false because for any ϵ>0 there exists some δ>0 that is small enough. It can't be any delta.
d.) for any $\epsilon \gt 0$, there exists $\delta \gt 0$ such that for all $x \in (0,1)$ and $0 \lt |x-x_0| \lt \delta$, one has $0 \lt |f(x) - f(x_0)|\leq \epsilon$.
this definition is false. I dont quite know how to provide an explanation for it tho.
these are my attempts on the identification. Could you please help me confirm these answers?
Thank you.