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 \(|f(x)-f(a)| \leq C|x-a|^{\gamma}\) for some \(\gamma>0\) and \(C>0\)
(a) If \(\gamma=1,\) show that f is continuous at a
(b) If \(\gamma>1,\) show that f is differentiable at a
Differentiability
Continuity
Limit
Sandwich Theorem
(a) We are given that \(|f(x)-f(a)| \leq C|x-a|\) for some \(C>0\).
We have to show that f is continuous at x=a . For this it's enough to show that \(\lim_{x\to a} f(x)=f(a)\).
\(|f(x)-f(a)| \leq C|x-a| \Rightarrow f(a)-C|x-a| \le f(x) \le f(a) + C|x-a| \)
Now taking limit \( x \to a\) we have , \( \lim_{x\to a} f(a)-C|x-a| \le \lim_{x\to a} f(x) \le \lim_{x\to a} f(a) + C|x-a| \)
Using Sandwich theorem we can say that \( \lim_{x\to a} f(x) = f(a) \) . Since \(\lim_{x\to a} -C|x-a| = \lim_{x\to a} C|x-a|=0 \)
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 \(\lim_{x\to a} \frac{f(x)-f(a)}{x-a} \) exists .
We are given that , \(|f(x)-f(a)| \leq C|x-a|^{\gamma}\) for some \(\gamma>1\) and \(C>0\) ,
which implies \( |\frac{f(x)-f(a)}{x-a} | \le C|x-a|^{\gamma -1} \)
\(\Rightarrow -C|x-a|^{\gamma -1} \le \frac{f(x)-f(a)}{x-a} \le C|x-a|^{\gamma -1} \)
Now taking \(\lim_{x\to a} \) we get by Sandwich theorem \(\lim_{x\to a}\frac{f(x)-f(a)}{x-a} =0 \) i.e f'(a)=0 .
Since , \( \lim_{x\to a} C|x-a|^{\gamma -1} = \lim_{x\to a} -C|x-a|^{\gamma -1} = 0 \) , for \( \gamma >1 \).
Hence f is differentiable at x=a proved .
\( f : R \to R \) be such that \( |f(x)-f(a)| \le k|x-y| \) for some \( k \in (0,1) \) and all \( x,y \in R \) . 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 \(|f(x)-f(a)| \leq C|x-a|^{\gamma}\) for some \(\gamma>0\) and \(C>0\)
(a) If \(\gamma=1,\) show that f is continuous at a
(b) If \(\gamma>1,\) show that f is differentiable at a
Differentiability
Continuity
Limit
Sandwich Theorem
(a) We are given that \(|f(x)-f(a)| \leq C|x-a|\) for some \(C>0\).
We have to show that f is continuous at x=a . For this it's enough to show that \(\lim_{x\to a} f(x)=f(a)\).
\(|f(x)-f(a)| \leq C|x-a| \Rightarrow f(a)-C|x-a| \le f(x) \le f(a) + C|x-a| \)
Now taking limit \( x \to a\) we have , \( \lim_{x\to a} f(a)-C|x-a| \le \lim_{x\to a} f(x) \le \lim_{x\to a} f(a) + C|x-a| \)
Using Sandwich theorem we can say that \( \lim_{x\to a} f(x) = f(a) \) . Since \(\lim_{x\to a} -C|x-a| = \lim_{x\to a} C|x-a|=0 \)
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 \(\lim_{x\to a} \frac{f(x)-f(a)}{x-a} \) exists .
We are given that , \(|f(x)-f(a)| \leq C|x-a|^{\gamma}\) for some \(\gamma>1\) and \(C>0\) ,
which implies \( |\frac{f(x)-f(a)}{x-a} | \le C|x-a|^{\gamma -1} \)
\(\Rightarrow -C|x-a|^{\gamma -1} \le \frac{f(x)-f(a)}{x-a} \le C|x-a|^{\gamma -1} \)
Now taking \(\lim_{x\to a} \) we get by Sandwich theorem \(\lim_{x\to a}\frac{f(x)-f(a)}{x-a} =0 \) i.e f'(a)=0 .
Since , \( \lim_{x\to a} C|x-a|^{\gamma -1} = \lim_{x\to a} -C|x-a|^{\gamma -1} = 0 \) , for \( \gamma >1 \).
Hence f is differentiable at x=a proved .
\( f : R \to R \) be such that \( |f(x)-f(a)| \le k|x-y| \) for some \( k \in (0,1) \) and all \( x,y \in R \) . Show that f must have a unique fixed point .
