Results 1 to 2 of 2

Thread: [SOLVED] Prove: if f(A) open subset of R the R is continuous

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    103

    [SOLVED] Prove: if f(A) open subset of R the R is continuous

    Let A be an open subset of $\displaystyle \Re$ and let $\displaystyle f: A \rightarrow \Re$ be an increasing function. Prove that if f(A) is an open subset of $\displaystyle \Re$ then f is continuous.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Amanda1990 View Post
    Let A be an open subset of $\displaystyle \Re$ and let $\displaystyle f: A \rightarrow \Re$ be an increasing function. Prove that if f(A) is an open subset of $\displaystyle \Re$ then f is continuous.
    Set $\displaystyle B=f(A)$, we are told that $\displaystyle A\text{ and }B$ are open subsets of $\displaystyle \mathbb{R}$.

    Let $\displaystyle a\in A$, so $\displaystyle f(a) \in B$ but $\displaystyle B$ is open so there exists $\displaystyle r>0$ so that $\displaystyle (f(a)-r,f(a)+r) \subseteq B$. From here and on we will prove that the $\displaystyle \delta,\epsilon$ definition of continuity is satisfied for all $\displaystyle \epsilon < r$. Since $\displaystyle f(a) - \epsilon,f(a)+\epsilon \in B$ it means there exists $\displaystyle a_1\in A \text{ and }a_2\in A$ so that $\displaystyle f(a_1) = f(a) - \epsilon \text{ and }f(a_2) = f(a) + \epsilon$. Now if $\displaystyle a_1 < x < a_2 \implies f(a_1) < f(x) < f(a_2)$ and so $\displaystyle f(a) - \epsilon < f(x) < f(a) + \epsilon \implies |f(x) - f(a)| < \epsilon$. Certainly, $\displaystyle a_1 < a < a_2$, so the neighborhood $\displaystyle N = \{ x\in \mathbb{R} | a_1 < x < a_2 \}$ of $\displaystyle a$ gets mapped within $\displaystyle \epsilon$ of $\displaystyle f(a)$. Thus, $\displaystyle f$ is continous at any point $\displaystyle a\in A$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving open subset
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: Apr 14th 2011, 10:10 PM
  2. Replies: 5
    Last Post: Apr 12th 2010, 02:16 PM
  3. Inverse of an open set is open implies continuous
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Sep 14th 2009, 05:15 AM
  4. is this subset of R2 closed or open
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 2nd 2008, 02:30 PM
  5. Replies: 5
    Last Post: Sep 9th 2007, 01:57 PM

Search Tags


/mathhelpforum @mathhelpforum