The Equivalence of the Epsilon - Delta and the Sequenctial Definitions of Continuity
Thedefinition of continuity states: If for a functionwiththen givensuch thatthere existssuch that ifthenthen we may sayis continuous at
The sequential definition states that if for all sequenceswithwe havethenis continuous at
We must prove these statements are equivalent.
To do this, suppose a functionwith domainis either continuous or not at according to thedefinition.
Suppose first thatdoes satisfy thedefinition of continuity atWe want to deduce that ifis any sequence inwiththen
Suppose thatis given . By assumption there issuch thatfor allwithand there is an integersuch thatfor all
hencefor allso that
Next suppose thatdoes not satisfy thedefinition atWe want to find a sequenceinwithbut
By assumption there is somesuch that for eachfor some
Applying this statement with.we findfor some with
The sequenceis inwithbut