f-x-p-q-x-p-q-p-Z-q-Z-q-0-p-q-1-x-10x-overtise-does-f-is-continuous-

Question Number 3655 by 123456 last updated on 17/Dec/15

f (x) = {\begin{cases} p + q & x = \frac{p}{q}, p \in Z, q \in Z, q \neq 0, (p, q) = 1 \\ ⌊ x ⌋ + ⌊ 10 x ⌋ & overtise \end{cases}

does f is continuous ?

Commented by prakash jain last updated on 18/Dec/15

c h o o s e p \notin Q, say p = π

Checking for definition

\forall ϵ > 0, \exists δ > 0 that if ∣ p - x ∣< ϵ then ∣ f (p) - f (x) ∣< δ

f (p) = 3 + 31 = 34

Choose ϵ = 1, for any δ we can choose x

3 + \frac{1}{34 + ⌊ δ + 1 ⌋} \Rightarrow∣ f (p) - f (x) ∣> δ

f (x) is not continous .