18. Let I := [a, b] and let f: I →ℝ be a (not necessarily continuous) function with the property

Name: 18. Let I := [a, b] and let f: I →ℝ be a (not necessarily continuous) function with the property that for every x ∈ I, the function f is bounded on a neighborhood V(x) of x (in the sense of Definition 4.2.1). Prove that f is bounded on I.
Uploaded: 2022-08-12T06:56:45-08:00
Duration: 3 min 21 s
Channel: Adi S
Description: 18. Let I := [a, b] and let f: I →ℝ be a (not necessarily continuous) function with the property that for every x ∈ I, the function f is bounded on a neighborhood V(x) of x (in the sense of Definition 4.2.1). Prove that f is bounded on I.

Question

18. Let $I := [a, b]$ and let $f: I \to \mathbb{R}$ be a (not necessarily continuous) function with the property that for every $x \in I$, the function $f$ is bounded on a neighborhood $V_{\delta_x}(x)$ of $x$ (in the sense of Definition 4.2.1). Prove that $f$ is bounded on $I$.

          18. Let $I := [a, b]$ and let $f: I \to \mathbb{R}$ be a (not necessarily continuous) function with the property that for every $x \in I$, the function $f$ is bounded on a neighborhood $V_{\delta_x}(x)$ of $x$ (in the sense of Definition 4.2.1). Prove that $f$ is bounded on $I$.

18. Let I := [a, b] and let f: I →ℝ be a (not necessarily continuous) function with the property that for every x ∈ I, the function f is bounded on a neighborhood V(x) of x (in the sense of Definition 4.2.1). Prove that f is bounded on I.

Added by Chris H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

08/12/2022

Step 1

Step 1: Since $f$ is bounded on a neighborhood of each point $x \in I$, we can find a neighborhood $V_{\delta_x}(x)$ of $x$ and a number $M_x > 0$ such that $|f(y)| \le M_x$ for all $y \in V_{\delta_x}(x)$. Show more…

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18. Let I:= [a, b] and let f: I → R be a (not necessarily continuous) function with the property that for every x ∈ I, the function f is bounded on a neighborhood V8x(x) of x (in the sense of Definition 4.2.1). Prove that f is bounded on I.