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Math Help - Discontinuous Functions

  1. #1
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    Discontinuous Functions

    Hey for some reason I'm really struggling with this one problem--

    Q: If f and g are both not continuous at a, then f+g is not continuous at a. Prove or Disprove.

    Any suggestions would be helpful!
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  2. #2
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    Quote Originally Posted by johnlaw View Post
    Hey for some reason I'm really struggling with this one problem--

    Q: If f and g are both not continuous at a, then f+g is not continuous at a. Prove or Disprove.

    Any suggestions would be helpful!
    How about taking a=0, ~ f(x) = \frac{1}{x}, ~ g(x) = -\frac{1}{x}?
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  3. #3
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    Let f(x) := -1 for x>0, 0 otherwise. Let g(x) := 1 for x>0, 0 otherwise.
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  4. #4
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    Quote Originally Posted by Defunkt View Post
    How about taking a=0, ~ f(x) = \frac{1}{x}, ~ g(x) = -\frac{1}{x}?
    A quick comment: for your example, f+g is not defined at 0
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  5. #5
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    Quote Originally Posted by FancyMouse View Post
    A quick comment: for your example, f+g is not defined at 0
    True, so define f(x)=\begin{cases} 0 & \mbox{if} \quad x=0 \\ \frac{1}{x} & \mbox{if} \quad x\ne0 \end{cases}. Then, f,-f are both not-continuous at zero but f+-f=0(x)
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