1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Continuous Functions

  1. Apr 14, 2012 #1
    Hi, can someone please check if my proof is correct

    1. a) Assume f : R -> R is continuous when the usual topology on R is
    used in the domain and the discrete topology on R is used in the range. Show
    that f must be a constant function.

    My attempt :

    Let f: R --> R be continuous. Suppose that f is not constant, then f assumes
    at least 2 values, p and q.
    In R we can find two disjoint open intervals I and J such that
    p is in I and q is in J.
    By continuity f^{-1} and f^{-1}[J] are open.
    They are disjoint as I and J are, and non-empty as p and q have preimages.
    This contradicts the fact above. So f must be constant.

    #1 b) Prove that for any two topological spaces (X; T1) and (Y; T2), if
    y0 is in Y then the constant function f : X -> Y defi ned by f(x) = y0 is continu-

    My Attempt:

    f:x->y is continuous if for every v in T2 , f inverse (v) is in T

    let y0 in Y then f(x) = y0 for every x in X is continuous and X is open set

    If y0 is not in v, then f inverse (v) = empty set so its always open....therefore every constant function is always continuous.

  2. jcsd
  3. Apr 14, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    Contradicts what fact?
    This is fine, but your write-up is sloppy.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook