There's something on my textbook that I don't understand:
Let $f:I \to \mathbb{R}$ be continuous on the interval $I$.So
$\forall x_0$ $\epsilon$ $I$ and $\forall \epsilon>0$ there $\exists \delta>0$ such that $\forall x$ $\epsilon$ $I$
$|x-x_0|<\delta$ $\implies$ $|f(x)-f(x_0)|<\epsilon$
It is obvious that in the definition above $\delta$ depends on $\epsilon$ and the point $x_0$ . If the interval $I$ is finite , then let $\delta_0>0$ be the smallest of all numbers $\delta>0$ such that :
$\forall x_0$ $\epsilon$ $I$ and $\forall x$ $\epsilon $ $I$ such that $|x-x_0|<\delta_0$ $\implies$ $|f(x)-f(x_0)|<\epsilon$
Up to this everything is clear.But then it states that:
If the interval $I$ is infinite , then the number $\delta_0>0$ doesn't always exist.
And then gives the definition of uniform continuity . I don't understand why in the second case the existence of the number $\delta_0>0$ is not assured,as I could understand it means that there can be cases when the smallest number $\delta>0$ with the property as mentioned above can not exist.How is this possible ? If anyone could give me an answer or a hint I would really appreciate it. Thank you!