GATE Mathematics Practice Quiz

Question 1

Let f:[0,1]	o\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]	o\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]	o\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
otin 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]	o\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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