The Equivalence of the Epsilon - Delta and the Sequenctial Definitions of Continuity
The
definition of continuity states: If for a function
with
then given
such that
there exists
such that if
thenthen we may say
is continuous at
The sequential definition states that if for all sequences
with
we have
thenis continuous at
We must prove these statements are equivalent.
To do this, suppose a functionwith domain
is either continuous or not at
according to thedefinition.
Suppose first thatdoes satisfy thedefinition of continuity atWe want to deduce that if
is any sequence inwith
then
Suppose thatis given . By assumption there issuch thatfor all
withand there is an integer
such that
for all
hence
for all
so that
Next suppose thatdoes not satisfy thedefinition at
We want to find a sequenceinwith
but
By assumption there is somesuch that for each
for some
Applying this statement with
.we findfor some
with
The sequence
is inwithbut
