0

Let $\lbrace f_n\rbrace$ and $\lbrace g_n\rbrace $ be two sequences of real-valued continuous functions on a compact subset of $\mathbb{R}$ such that they converge pointwise to $f$ and $g$ respectively, where $f$ and $g$ are continuous functions on $\mathbb{R}$. Then what can we say about the composite function $f_n \circ g_n$. Can we say that it will converge to $f\circ g$.

A similar question is asked here Convergence of composition of functions sequences. But here the continuity of functions $f$ and $g$ was not given. Please at least give me some hint to solve this.

MANI
  • 1,928
  • 1
  • 11
  • 23
  • Please see if my revised answer is OK. – Kavi Rama Murthy Aug 31 '19 at 13:43
  • @KaviRamaMurthy, Thanku sir for your continuous support, I have a modification in the question that if the domain of the sequence of functions $\lbrace f_n \rbrace$ and $ \lbrace g_n \rbrace $ is compact then what we can say about the convergence of the composition $f_n \circ g_n$? – MANI Sep 02 '19 at 05:31

2 Answers2

1

Let $f_n(x)=nx$ for $0 \leq x \leq 1/n$, $2-nx$ for $ 1/n \leq x \leq 2/n$ and $0$ for $x >2/n$. $g_n(x)=\frac 1 n$. Let $f(x)=0$ for all $x$ and $g(x)=0$ for all $x$. Then $f_n(g_n(x))\to 1\neq f(g(x))$.

Kavi Rama Murthy
  • 311,013
  • Thanks for your answer sir. – MANI Aug 30 '19 at 12:41
  • Ram Murthy, here $f$ is discontinuous sir but in the given question $f$ should be continuous. – MANI Aug 31 '19 at 12:56
  • I have edited the answer. Now the limiting functions are both $0$. – Kavi Rama Murthy Aug 31 '19 at 13:24
  • Ram Murthy, yes sir this will work for the above. Thanks for your continuous support through this platform. – MANI Aug 31 '19 at 15:27
  • @Rahul Maurya Please watch the above answer, I think you will understand it easily. – MANI Aug 31 '19 at 15:28
  • sir, what can we say about this- Let $ \lbrace f_n\rbrace$ and $\lbrace g_n\rbrace $ be two sequences of real-valued continuous functions on compact subset of R such that they converge point wise to $f$ and $g$ respectively, where $f$ and $g$ are continuous functions on R. Then what can we say about the composite function $f_n \circ g_n$. Can we say that it will converge to f∘g? – MANI Sep 02 '19 at 05:20
  • @MANISHANKARPANDEY $f_n\circ g_n$ has to be defined for the question to make sense. – Kavi Rama Murthy Sep 02 '19 at 05:26
  • Suppose it is defined then what we can say, actually Sir I am working on a problem of functional analysis and got stuck at this point. – MANI Sep 02 '19 at 05:36
-1

Let $\lbrace f_n\rbrace$ and $\lbrace g_n\rbrace $ be two sequences of real-valued continuous functions on compact subset of $\mathbb{R}$ such that they converge pointwise to $f$ and $g$ respectively, where $f$ and $g$ are continuous functions on $\mathbb{R}$. Then what can we say about the composite function $f_n \circ g_n$. Can we say that it will converge to $f\circ g$?

rahul maurya
  • 1