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Continuity at a particular x

  1. Jul 27, 2008 #1
    Can anyone help me with this problem?

    Say f(x) & g(x) are cont. at x=5.
    Also that f(5)=g(5)=8.

    If h(x)=f(x) when x<=5
    h(x)=g(x) when x>=5:

    prove h(x) is cont at x=5.
  2. jcsd
  3. Jul 27, 2008 #2
    Hmm, this appears to be a straightforward application of the definition of continuity at a point. You already know the h(5) exists, now you just need to show that the limit as x approaches 5 of h(x) is 8. This can be done as 2 particular cases, as h(x) either =f(x) or g(x), depending on which side you approach the point from.
  4. Jul 27, 2008 #3


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    Welcome to PF!

    Hi blackisback ! Welcome to PF! :smile:

    Just write out the definitions of:

    f(x) is continuous at x = 5
    g(x) is continuous at x = 5
    h(x) is continuous at x = 5

    using the δ and ε method.

    Then just chug away. :smile:
  5. Jul 27, 2008 #4
    thanks alot guys
  6. Jul 28, 2008 #5


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    I don't think you need to use "epsilon-delta". Just using the one-sided limits should be enough.
  7. Aug 2, 2008 #6
    Is 5 an accumulation point, ???
  8. Aug 3, 2008 #7
    Since the function is continuous there, 5 is an accumulation point of the image of h under the standard topology inherited from R.
  9. Aug 3, 2008 #8
    Since all functions are continuous over an isolated point in there domain your implication is incorrect
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