This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 2 based on use of Sandwich Theorem . Let's give it a try !!
Let f be a real valued function satisfying 
for some 
and 
(a) If 
show that f is continuous at a
(b) If 
show that f is differentiable at a
Differentiability
Continuity
Limit
Sandwich Theorem
(a) We are given that 
for some .
We have to show that f is continuous at x=a . For this it's enough to show that 
.
Now taking limit 
we have , 
Using Sandwich theorem we can say that 
. Since 
Hence f is continuous at x=a proved .
(b) Here we have to show that f is differentiable at x=a for this it's enough to show that the 
exists .
We are given that , for some 
and ,
which implies 
Now taking 
we get by Sandwich theorem 
i.e f'(a)=0 .
Since , 
, for 
.
Hence f is differentiable at x=a proved .
be such that 
for some 
and all 
. Show that f must have a unique fixed point .
This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 2 based on use of Sandwich Theorem . Let's give it a try !!
Let f be a real valued function satisfying for some and
(a) If show that f is continuous at a
(b) If show that f is differentiable at a
Differentiability
Continuity
Limit
Sandwich Theorem
(a) We are given that for some .
We have to show that f is continuous at x=a . For this it's enough to show that .
Now taking limit we have ,
Using Sandwich theorem we can say that . Since
Hence f is continuous at x=a proved .
(b) Here we have to show that f is differentiable at x=a for this it's enough to show that the exists .
We are given that , for some and ,
which implies
Now taking we get by Sandwich theorem i.e f'(a)=0 .
Since , , for .
Hence f is differentiable at x=a proved .
be such that for some and all . Show that f must have a unique fixed point .
