Explain why this is no good as a definition of continuity

  • Thread starter gregy6196
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Explain why this is no good as a definition of continuity at a point a (either by giving an example of a continuous function that does not satisfy the definition or a discontinuous one that does):
Given ε > 0 there exists a [itex]\delta[/itex] > 0 such that |x – a| < [itex]\epsilon[/itex] [itex]\Rightarrow[/itex] |f(x) – f(a)| < [itex]\delta[/itex]
 

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  • #2
mathman
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There will always exist δ. For example f(x)=0 for x< 0 and f(x) = 1 otherwise. Then your definition will have continuity at 0 using δ > 1.
 

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