1.Let f,g: I R be continuous at . Show that the product fg is contiuous at E.Here you may either prove it by definition or prove it via the following theorem that we proved in class(Theorem 2.1): h: I → R is conitinuous at & E I if and only if the following holds true: for anysequence {In} in I such that lim an = , it holds that lim f(n) = f().%3D

Theorem: Let f: I→ℝ be function. f is continuous at ε∈I if and only if for any sequence xn in I such…

Answered: 1. Let f,g: I R be continuous at . Show…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1.
Let f,g: I R be continuous at . Show that the product fg is contiuous at E.
Here you may either prove it by definition or prove it via the following theorem that we proved in class
(Theorem 2.1): h: I → R is conitinuous at & E I if and only if the following holds true: for any
sequence {In} in I such that lim an = , it holds that lim f(n) = f().
%3D

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Step 1

Theorem: Let $f : I \to ℝ$ be function. $f$ is continuous at $ε \in I$ if and only if for any sequence $(x_{n})$ in $I$ such that $\lim_{n \to \infty} x_{n} = ε$ , it holds that $\lim_{n \to \infty} f (x_{n}) = f (ε)$ .

It is given that, $f, g : I \to ℝ$ is continuous at $ε$ .

Then by the definition, for every sequence $(x_{n})$ in $I$ such that $\lim_{n \to \infty} x_{n} = ε$ , it holds that $\lim_{n \to \infty} f (x_{n}) = f (ε)$ and $\lim_{n \to \infty} g (x_{n}) = f (ε)$ .

To prove $f g$ is continuous at $ε$ , it is enough to prove for every sequence $(x_{n})$ in $I$ such that $\lim_{n \to \infty} x_{n} = ε$ , it holds that $\lim_{n \to \infty} f g (x_{n}) = f g (ε)$ .

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