This is a problem from ISI MStat 2018 PSA Problem 7 based on Continuity.
Let be a function defined from
to
such that
and
for all x
Then is
Epsilon-Delta definition of limit
Continuity
Bounded function
Answer: is (A)
ISI MStat 2018 PSA Problem 7
Introduction to Real Analysis by Bertle Sherbert
Try to use the epsilon-delta definition of limit and the property that for all x.
such that
. Now
for all
for every
.
Let's try to use this .
Given any we can select a suitable
such that
. Then
. But
. Hence ,
. Hence , for all
we have
. Since ,
is arbitrary , we must have
for all
So is continuous and bounded
This is a problem from ISI MStat 2018 PSA Problem 7 based on Continuity.
Let be a function defined from
to
such that
and
for all x
Then is
Epsilon-Delta definition of limit
Continuity
Bounded function
Answer: is (A)
ISI MStat 2018 PSA Problem 7
Introduction to Real Analysis by Bertle Sherbert
Try to use the epsilon-delta definition of limit and the property that for all x.
such that
. Now
for all
for every
.
Let's try to use this .
Given any we can select a suitable
such that
. Then
. But
. Hence ,
. Hence , for all
we have
. Since ,
is arbitrary , we must have
for all
So is continuous and bounded