Q. 66 Prove that if $f$ is continuou... [FREE SOLUTION]

Chapter 4: Q. 66 (page 401)

Prove that if $f$ is continuous on $[a, b]$ and we define $F(x)=$ $\int_{a}^{x} f(t) d t$, then $F$ is continuous on the closed interval $[a, b]$. (Hint: Argue that $F$ is continuous on $(a, b)$, left-continuous at $a$, and right-continuous at b.)

Short Answer

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\text { The functions } A(x)=\int_{a}^{x} f(t) d t \text { and } B(x)=\int_{b}^{x} f(t) d t \text { differ by a constant. }

Step by step solution

Given information

The objective is to prove that for all real numbers $a, b$, the functions $A(x)=\int_{a}^{x} f(t) d t$ and $B(x)=\int_{b}^{x} f(t) d t$ differ by a constant.

Calculation

Since $f$ is continuous on all $\mathbb{R}$.

$A(x)-B(x)$ $=\int_{a}^{x} f(t) d t-\left(\int_{b}^{x} f(t) d t\right)$ $=\int_{a}^{x} f(t) d t-\int_{x}^{b} f(t) d t \quad$ [fundamental property of calculus] $=\int_{a}^{b} f(t) d t$ $=F(b)-F(a)$

The expression $F(b)-F(a)$ is a constant.

Therefore, the functions $A(x)=\int_{a}^{x} f(t) d t$ and $B(x)=\int_{b}^{x} f(t) d t$ differ by a constant.

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91影视

Prove that if \(f\) is continuous on \([a, b]\) and we define \(F(x)=\) \(\int_{a}^{x} f(t) d t\), then \(F\) is continuous on the closed interval \([a, b]\). (Hint: Argue that \(F\) is continuous on \((a, b)\), left-continuous at \(a\), and right-continuous at b.)

Short Answer

Step by step solution

Given information

Calculation

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