But whatwe actuallyhaveis the originalfunction, that we knowis not defined
at At we havea hole in the graph,or a discontinuityof the functionat That
is, the functionis definedfor all otherx-valuescloseto
Looselyspeaking,if we wereto hand-drawthe graph,we wouldneedto take our penciloff the pagewhen
we got to this hole,leavinga gap in the graphas indicated:
Now we will formalizethe propertyof continuityof a functionand providea test for determiningwhenwe
havecontinuousfunctions.
Continuityof a Function
Definition: The functionf(x) is continuousat x = aif the followingconditionsall hold:
is in the domainoff(x);
exists;
Notethat it is possibleto havefunctionswheretwo of theseconditionsare satisfiedbut the third is not.
Considerthe piecewisefunction