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Show that if f: R→ R is continuous and f(x) = 0 for every rational number a, men f(x)= 0 for all z.

Show that if f: R→ R is continuous and f(x) = 0 for every rational number a, men f(x)= 0 for all z.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 50E
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**Exercise 6:** Show that if \( f : \mathbb{R} \rightarrow \mathbb{R} \) is continuous and \( f(x) = 0 \) for every rational number \( x \), then \( f(x) = 0 \) for all \( x \). **Exercise 7:** Suppose that the function \( f : [a, b] \rightarrow \mathbb{R} \) is continuous. For any natural number \( k \), let \( x_1, x_2, \ldots, x_k \) be points in \([a, b]\). Prove that there is a point \( z \) in \([a, b]\) such that \[ f(z) = \frac{f(x_1) + \ldots + f(x_k)}{k}. \]
Transcribed Image Text:**Exercise 6:** Show that if \( f : \mathbb{R} \rightarrow \mathbb{R} \) is continuous and \( f(x) = 0 \) for every rational number \( x \), then \( f(x) = 0 \) for all \( x \). **Exercise 7:** Suppose that the function \( f : [a, b] \rightarrow \mathbb{R} \) is continuous. For any natural number \( k \), let \( x_1, x_2, \ldots, x_k \) be points in \([a, b]\). Prove that there is a point \( z \) in \([a, b]\) such that \[ f(z) = \frac{f(x_1) + \ldots + f(x_k)}{k}. \]
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