GATE Mathematics Practice Quiz

Question 1

Let $f:[0,1]\to\mathbb{R}$ be defined by $f(x)=1$ if x is rational and $f(x)=0$ if x is irrational. Which statement is correct regarding Riemann integrability of f on [0,1]?

Accepted Answer

f is not Riemann integrable because it is discontinuous at every point of [0,1]

Question 2

Define $f:[0,1]\to\mathbb{R}$ by $f(x)=1$ if $x=1/n$ for some $n\in\mathbb{N}$, and $f(x)=0$ otherwise. Which of the following is true?

Accepted Answer

f is Riemann integrable and $\int_0^1 f(x)\,dx=0$

Question 3

Define $f:[0,1]\to\mathbb{R}$ by $f(x)=1/q$ if $x=p/q$ in lowest terms with $q\ge 1$, and $f(x)=0$ if x is irrational. Which statement is correct?

Accepted Answer

f is Riemann integrable and $\int_0^1 f(x)\,dx=0$

Question 4

Let $C\subset[0,1]$ be the Cantor set, and define $f(x)=1$ if $x\in C, f(x)=0$ if $x\notin C$. Which statement is correct?

Accepted Answer

f is Riemann integrable and $\int_0^1 f(x)\,dx=0$

Question 5

Let $f:[0,1]\to\mathbb{R}$ be defined by $f(x)=0$ for $x\in[0,1/2)$ and $f(x)=1$ for $x\in[1/2,1]$. Which statement is correct?

Accepted Answer

f is Riemann integrable and $\int_0^1 f(x)\,dx=1/2$

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