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Math Help - Cauchy definition (epsilon-delta) of continuous functions

  1. #1
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    Cauchy definition (epsilon-delta) of continuous functions

    Hi can someone please help with this question:

    Cauchy definition (epsilon-delta) of continuous functions-1.jpg

    What i Know

    I know that this graph is not continuous at a=0 but i am finding it difficult to prove this using Cauchy definition (epsilon-delta) of continuous functions.

    Thanks Subzero
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  2. #2
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    Quote Originally Posted by SubZero View Post
    Hi can someone please help with this question:

    Click image for larger version. 

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    What i Know

    I know that this graph is not continuous at a=0 but i am finding it difficult to prove this using Cauchy definition (epsilon-delta) of continuous functions.

    Thanks Subzero
    Good! You are probably finding it difficult to show that function is not continous at 0 because it is continuous at 0!

    Given any \epsilon> 0 if x> 0, f(x)= x so we can just take \delta= \epsilon. That way, "if |x- 0|= |x|> \delta, |f(x)- f(0)|= |x- 0|= |x|< \delta= \epsilon". While if x< 0, f(x)= 0 so |f(x)- f(0)|= |0- 0|= 0< \epsilon for any x and so for |x- 0|= |x|< \delta.

    (The function is not differentiable at x= 0 but is continuous there.)
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  3. #3
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    Thanks for the help
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