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L8 Real Analysis II Theorem 1: L(P1,f)≤U(P2,f). Theorem 2: If 𝑃⊆𝑃*, 𝑈(𝑃*,𝑓)−𝐿(𝑃*,𝑓)≤𝑈(𝑃,𝑓)−𝐿(𝑃,𝑓)

L8 Real Analysis II Theorem 1: L(P1,f)≤U(P2,f). Theorem 2: If 𝑃⊆𝑃*, 𝑈(𝑃*,𝑓)−𝐿(𝑃*,𝑓)≤𝑈(𝑃,𝑓)−𝐿(𝑃,𝑓) EasyMath with Dr. Rajashree Basavaraj

48 views - 10 months ago

L19 Real Analysis II If f∈R(α), then f^2∈R(α).

L19 Real Analysis II If f∈R(α), then f^2∈R(α). EasyMath with Dr. Rajashree Basavaraj

26 views - 10 months ago

L2 Partition, Riemann Stieltjes integral, Upper & Lower Sum & Upper & Lower RSI, Norm & Refinement.

L2 Partition, Riemann Stieltjes integral, Upper & Lower Sum & Upper & Lower RSI, Norm & Refinement. EasyMath with Dr. Rajashree Basavaraj

102 views - 11 months ago

L13 Real Analysis II If f∈R(α) then m(α(b)−α(a))≤ ∫fdα ≤M(α(b)−α(a)).

L13 Real Analysis II If f∈R(α) then m(α(b)−α(a))≤ ∫fdα ≤M(α(b)−α(a)). EasyMath with Dr. Rajashree Basavaraj

64 views - 10 months ago

L1 Real analysis II. Upper & Lower bound, Supremum & Infimum, Monotonic function & Bounded function

L1 Real analysis II. Upper & Lower bound, Supremum & Infimum, Monotonic function & Bounded function EasyMath with Dr. Rajashree Basavaraj

221 views - 11 months ago

L18 Real Analysis II If f∈R(α) on [a,b], then ∣f∣∈R(α) on [a,b] and | ∫fdα| = ∫ ∣f∣ dα.

L18 Real Analysis II If f∈R(α) on [a,b], then ∣f∣∈R(α) on [a,b] and | ∫fdα| = ∫ ∣f∣ dα. EasyMath with Dr. Rajashree Basavaraj

80 views - 10 months ago

L21 Real Analysis II First Mean Value Theorem for the Riemann–Stieltjes integral.

L21 Real Analysis II First Mean Value Theorem for the Riemann–Stieltjes integral. EasyMath with Dr. Rajashree Basavaraj

305 views - 10 months ago

L11 Real Analysis II. If f is monotonic on [a, b] and α is continuous on [a, b] then f∈R(α).

L11 Real Analysis II. If f is monotonic on [a, b] and α is continuous on [a, b] then f∈R(α). EasyMath with Dr. Rajashree Basavaraj

304 views - 10 months ago

L15 RA II If f,g∈R(α) & f(x)≤g(x) then ∫fdα ≤ ∫gdα. If f∈R(α) & ∣f(x)∣≤k then |∫fdα| ≤k(α(b)−α(a)).

L15 RA II If f,g∈R(α) & f(x)≤g(x) then ∫fdα ≤ ∫gdα. If f∈R(α) & ∣f(x)∣≤k then |∫fdα| ≤k(α(b)−α(a)). EasyMath with Dr. Rajashree Basavaraj

63 views - 10 months ago

L7 Real Analysis II

L7 Real Analysis II EasyMath with Dr. Rajashree Basavaraj

38 views - 11 months ago

L12 Real Analysis II If f∈R(α) and g∈R(α), then f+g∈R(α) on [a,b].

L12 Real Analysis II If f∈R(α) and g∈R(α), then f+g∈R(α) on [a,b]. EasyMath with Dr. Rajashree Basavaraj

128 views - 10 months ago

L4 Real Analysis II. If bounded function f:[a,b]→R, Lower Reimann Integral ≤ Upper Reimann Integral.

L4 Real Analysis II. If bounded function f:[a,b]→R, Lower Reimann Integral ≤ Upper Reimann Integral. EasyMath with Dr. Rajashree Basavaraj

88 views - 11 months ago

L9 Real Analysis II Theorem: If f∈R(α) on [a,b], and c∈(a,b), then f∈R(α) on [a,c], and [c,b]

L9 Real Analysis II Theorem: If f∈R(α) on [a,b], and c∈(a,b), then f∈R(α) on [a,c], and [c,b] EasyMath with Dr. Rajashree Basavaraj

37 views - 10 months ago

L17 Real Analysis II If f∈R(α1) and f∈R(α2) then f∈R(α1+α2), and ∫fd(α1+α2) = ∫fdα1 + ∫fdα2.

L17 Real Analysis II If f∈R(α1) and f∈R(α2) then f∈R(α1+α2), and ∫fd(α1+α2) = ∫fdα1 + ∫fdα2. EasyMath with Dr. Rajashree Basavaraj

41 views - 10 months ago

L10 Real Analysis II If f is continuous on [a, b], then f is Riemann-Stieltjes integrable on [a, b].

L10 Real Analysis II If f is continuous on [a, b], then f is Riemann-Stieltjes integrable on [a, b]. EasyMath with Dr. Rajashree Basavaraj

195 views - 10 months ago

L5 Real Analysis II. Condition of integrabilty. Riemann-Stieltjes integrabilty.

L5 Real Analysis II. Condition of integrabilty. Riemann-Stieltjes integrabilty. EasyMath with Dr. Rajashree Basavaraj

56 views - 11 months ago

L6 Real Analysis II

L6 Real Analysis II EasyMath with Dr. Rajashree Basavaraj

44 views - 11 months ago

L20 Real Analysis II If f∈R(α), then f∈R(cα). Also ∫ f(x)d(cα(x))=c∫f(x)dα(x).

L20 Real Analysis II If f∈R(α), then f∈R(cα). Also ∫ f(x)d(cα(x))=c∫f(x)dα(x). EasyMath with Dr. Rajashree Basavaraj

35 views - 10 months ago

L16 Real Analysis II If f∈R(α) on [a,b], then cf∈R(α) and ∫cf(x)dα=c∫f(x)dα for any constant c∈R.

L16 Real Analysis II If f∈R(α) on [a,b], then cf∈R(α) and ∫cf(x)dα=c∫f(x)dα for any constant c∈R. EasyMath with Dr. Rajashree Basavaraj

57 views - 10 months ago

L22 Real Analysis II Second mean value theorem for Riemann Stieltjes Integral.

L22 Real Analysis II Second mean value theorem for Riemann Stieltjes Integral. EasyMath with Dr. Rajashree Basavaraj

301 views - 10 months ago