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Math Help - Continuity (Epsilon-Delta)

  1. #1
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    Continuity (Epsilon-Delta)

    How would i approach a question that asks me to use the Cauchy Definition (Epsilon-Delta) to prove that f(x) = x is continuous at x = 1.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    What is wanted : \forall \varepsilon>0,\ \exists \ \delta \ | \ |x-1|<\delta \Rightarrow |f(x)-1|=|x-1|<\varepsilon.

    The aim is to show that there exists such a \delta for each \varepsilon. ( \delta will depend on \varepsilon but can't depend on x). It can be achieved by taking \varepsilon>0, and looking for a condition on |x-1| so that |f(x)-1|<\varepsilon.

    An example :

    Let's show the continuity of x \mapsto x^2 at 0.

    What is wanted : \forall \varepsilon>0,\ \exists \ \delta \ | \ |x-0|<\delta \Rightarrow |x^2-0|<\varepsilon which can be rewritten \forall \varepsilon>0,\ \exists \ \delta \ | \ |x|<\delta \Rightarrow x^2<\varepsilon \ (1)

    What condition has to be respected by x so that x^2<\varepsilon ? Take the square root of this inequality ( x\mapsto\sqrt{x} being an increasing function) : x^2<\varepsilon \Leftrightarrow x<\sqrt{\varepsilon} and here is the condition. Hence, by taking \delta=\sqrt{\varepsilon}, (1) is true and we've shown the continuity of the function at 0.
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