Question

Every bounded monotonic function on [a, b] is.
R-
integrab
Not R-
integrable
Partially R-
integrable
None of​

Answer 1

(b) Not R- intergrable.

Not every bounded function is integrable. For example the function f(x)=1 if x is rational and 0 otherwise is not integrable over any interval [a, b] (Check this). In general, determining whether a bounded function on [a, b] is integrable, using the definition, is difficult.

A function is Riemann integrable on [a,b] iff it is bounded and continuous save on a set of Lebesgue measure zero. Here the discontinuities of f are within the finite set {a,b} and so f is Riemann integrable.

Answer 2

The correct answer is option (b) Not R- intergrable.

Explanation:

• Every bounded monotonic function on [a, b] is Not R- intergrable.
• Its true that Not every bounded function is integrable.
• For example the function f(x)=2 if x is rational and 0 otherwise is not integrable over any interval [a, b] (Check this).
• In general, determining whether a bounded function on [a, b] is integrable, using the definition, is difficult.
• A function is Riemann integrable on [a,b] if it is bounded and continuous save on a set of Lebesgue measure zero.
• Here the discontinuities of f are within the finite set {a,b} and so f is Riemann integrable.