Measure Theory and Integration-II Quiz

Every ........................ integral is absolutely continuous.

• Riemann
• continuous

• Borel
• indefinite

The value of r⁺ = r if ........

• r is a positive number
• r is a complex number.
• r is equal to zero.
• r is a negative number.

An absolutely continuous function is ....................

• bounded

• indefinite

• constant

• continuous

If f is of bounded variation on [a, b], then f'(x)

• is differentiable on [a, b].
• exists almost everywhere on [a, b]
• is continuous on [a, b].
• is constant on [a, b].

If f is absolutely continuous on [a, b] and f'(x) = 0 a.e then f is .....................

• f is constant
• f is discontinuous.
• f is decreasing.
• f is increasing.

When we say that a function f is of bounded variation?

• A function f is of bounded variation if it has a finite total variation over a given interval.
• A function f is of bounded variation if it has a finite limit at every point over a given interval.
• A function f is of bounded variation if it has a constant value over a given interval.
• A function f is of bounded variation if it has a continuous derivative over a given interval.

A monotone function f on [a, b] is called .................... if f' = 0 a.e.

• decreasing
• singular

• increasing
• non-monotonic

If f is of bounded variation on [a, b] then T_a^b= ................

• P_a^b

• N_a^b

• P_a^b+N_a^b

• 0

Say true or false: If f is absolutely continuous then f has a derivative almost everywhere.

• True
• Only if f is continuous.
• Not enough information to determine.
• False.

Say true or false: Every absolutely continuous functions is the definite integral of its derivative.

• False.
• True