- #1
iloveannaw
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Homework Statement
[tex]f: ]0, \infty[ \rightarrow \mathbb{R} [/tex] is defined as
[tex]f = 0[/tex] if x is irrational and
[tex]f = \frac{1}{n} [/tex]
if [tex] x = \frac{m}{n}[/tex]
where m and n are co-prime
Show that f is only then continuous about x0 when [tex]x \in \frac{\mathbb{R}}{\mathbb{Q}}[/tex]
The Attempt at a Solution
with [tex] x, y \in \mathbb{Q}, f\left(x\right) = \frac{1}{n}[/tex]
with [tex]\left|x - y\right| < \delta [/tex]
[tex]\forall x,y \in \mathbb{Q} \exists z [/tex] such that
[tex]x < z < y[/tex] or [tex]x > z > y[/tex]
such that
[tex]f\left(z\right) = 0[/tex] ,e.g. z is irrational
with the result that
[tex]\left|x - z\right| < \left|x - y\right| < \delta [/tex]
and
[tex]\left|f\left(x\right) - f\left(z\right)\right| > \left|f\left(x\right) - f\left(y\right)\right| < \epsilon [/tex]
for all [tex]x \in ]0, \infty[[/tex]
is this on the right lines?