Question

सिद्ध करो कि प्रत्येक एकदिष्ट फल्स रीमान समाकालनीय होता हैं।
Prove that every monotonic function is Riemann-integrable.​

Answer 1

Explanation:

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Answer 2

Explanation is given below.

Explanation:

• The function f is claimed to be Riemann integrable if its lower and upper integral are an equivalent. When this happens we define ∫f(x)dx=L(f,a,b)=U(f,a,b).
• Let f be a monotone function on [a, b] then f is integrated on [a, b].
• Evidence of increasing functions is equivalent. First note that if f decreases monotonically then f(b) ≤ f(x) ≤ f(a) for all x [a, b] then f is bounded to [a, b].
• Checking for monotonic functions states: Suppose a function is continuous on [a, b] and can be distinguished on [a, b] (a, b).
• If the derivative is greater than zero for all x in (a, b), the function will be increased to [a, b].
• If the derivative is less than zero for all x in (a, b), the function will decrease to [a, b].
• Not all bound function can be combined. For example, the function f(x)=1 if x is rational and 0 otherwise cannot be integrated over any interval [a, b].
• In general, it is difficult to decide if the bounded function on [a, b] can be integrated with the description.