- #1

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Let [tex]f:\mathbb{R}\rightarrow \mathbb{R}[/tex]. The function [tex]f[/tex] is said to be continuous at the point [tex]a\in \mathbb{R}[/tex], if given [tex]\epsilon > 0[/tex], there is a [tex]\delta > 0[/tex], such that

[tex]|f(x)-f(a)|<\epsilon[/tex],

whenever

[tex]|x-a|<\delta[/tex].

The function [tex]f[/tex] is said to be continuous if it is continuous at each point of [tex]\mathbb{R}[/tex].

I'm confused because, if I understand correctly, we can set both [tex]\epsilon[/tex] and [tex]\delta[/tex] to be any numbers. Consider for example this function:

[tex]f(x) =\begin{cases}

1 & \text{ if } floor(x) \text { is odd } \\

2 & \text{ if } floor(x) \text{ is even }

\end{cases}[/tex]

With 0 considered even. If we let [tex]\epsilon = 98^{8000}[/tex], then this function is continuous, as all [tex]f(x)[/tex] are within [tex]\epsilon[/tex] of each other.

So what the heck, man? Is this a "weak" definition? Can a function be "continous" even if it is disconnected? What am I misunderstanding?