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Math Help - Proving a function is discontinuous

  1. #1
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    Proving a function is discontinuous

    My task is to show that the following function is discontinuous. <br />
f(x) = 1 \, \text{for } x>0 \wedge f(x) = 0 \, \text{for } x \leq 0, x_0 =0.
    Thanks in advance for your help.
    -The Doctor
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  2. #2
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    We say a function is continuous at a point x_0 if

    \underset{\epsilon>0}{\forall} \; \underset{\delta>0}{\exists} \; \underset{x\in \mathbb R}{\forall} \; |x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\epsilon

    Thus to prove your function (called the Heaviside function and usually noted H) is discontinuous at x_0=0, we must prove that

    \underset{\epsilon >0}{\exists} \; \underset{\delta >0}{\forall} \; \underset{x\in \mathbb R}{\exists} \; |x|<\delta \wedge |f(x)|>\epsilon (since x_0=f(x_0)=0)

    Choose \epsilon = \frac{1}{2}, and let \delta >0. We then have

    |\frac{\delta}{2}|=\frac{\delta}{2}<\delta

    but also

    |f(\frac{\delta}{2})|=|1-0|=1>\frac{1}{2}=\epsilon,

    so H is discontinuous at x_0=0.
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  3. #3
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    Thanks so much for your help. I really appreciated it.
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