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First let's state the $ε-δ$ defi nition of continuity of $f$ at the point $x = a$. Then here is the problems: (a) Prove that every bounded open (resp. closed) interval is an open (resp. a closed) ball.

(b) Continuing with the notation of the $ε-δ$ defi nition of continuity of $f$ at the point $x = a$. Say that $f$ is $b$-continuous at $x = a$ if, for every $s > 0$ there is an $r > 0$ such that if $x ∈B_{r}^{o}(a)$, then $f(x) ∈ B_{s}^{o}(b)$ where $b = f(a)$. Prove that $f$ is $b$-continuous at $x = a$ i ff f is continuous at $x = a$.

(c) Continuing with the notation of part (b). Say that $f$ is $o$-continuous at $x = a$ if, for every open set $V$, $b ∈V$, there is an open set $U$, $ a∈U $such that $x ∈ U$ implies $f(x) ∈ V$ . Prove that $f$ is $o$-continuous at $x = a$ iff $f$ is continuous at $x = a$.

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what are you having difficulties with? –  Ittay Weiss Jan 21 '13 at 1:44
    
You say "let's state the $\varepsilon$-$\delta$ definition of $f$ at the point $x=a$" and yet you never state the definition...? –  Clayton Jan 21 '13 at 1:54

1 Answer 1

For part (a), just take the midpoint of the interval and let the radius of the ball be the length of the interval divided by $2$.

What have you tried for parts (b) and (c)?

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