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The epsilon delta Criterion for Continuity

A function is continuous if it has no 'gaps' in it. Thecriterion states this in a rigorous form which makes it possible to prove that a function is continuous or not.


The criterion states:

A functionis continuous at a pointif forgiven there existssuch thatIn this definitionis generally a function ofOften the function needs to be defined to an interval, so that x is restricted to a certain range, for example

The criterion is best illustrated with examples.


Example:

so ifthen

Putthen as required for continuity.


Example:

On the intervalsoso ifthen

Putthen as required for continuity.


Example:

On the intervalsoso ifthen

Putthen as required for continuity.

In this case the domain is important. The function would not be continuous if 0 were part of the domain sinceis not defined forso we can find no M such thatfor allsatisfying