/sci/ - Science & Math

>>15074015
Ah, sorry for the line break.
>You can't do [math]|f(x)−f(y)|<M|x−y|[/math], that's nonsense
The idea is that since (1) [math]f(x) \leq Mx[/math], I assume that [math]x > y[/math] so [math]|x - y| = x - y[/math], then I just take [math]f(x) - f(y) \leq Mx - My[/math] from (1). Then [math]Mx - My = M|x - y| < M \delta = \epsilon[/math].
I think the arithmetic here makes sense? The assumptions basically make everything positive, so the modulus doesn't really matter, the non-decreasing assumption also means that [math]x > y \implies |f(x) - f(y)| = f(x) - f(y)[/math]
I think I'm doing something wrong wrt the quantifiers in the definition of continuity but I honestly can't see it, despite the result being nonsense.