Show that if f = R[a, b] then = R[c, d] for all closed subintervals...

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Show that if f = R[a, b] then ƒ = R[c, d] for all closed subintervals [c, d] of [a, b]. Let P = {x} be in P[a, b]. Let f : [a,b] → R be such that f is bounded on [a, b] and constant on each open subinterval (xi-1, xi) - say, f(x) = y; for x;-1 < x < x; for i = 1, 2, ..., n. Such a function f is a step function. Show that f = R[a, b] and Sa f(x)dx = ΣyiAx¡. If ƒ € R[a, b], show that for f(x)dx = lime→a+ √ f(x)dx. Show that if f = R[a, b] then ƒ = R[c, d] for all closed subintervals [c, d] of [a, b]. Let P = {x} be in P[a, b]. Let f : [a,b] → R be such that f is bounded on [a, b] and constant on each open subinterval (xi-1, xi) - say, f(x) = y; for x;-1 < x < x; for i = 1, 2, ..., n. Such a function f is a step function. Show that f = R[a, b] and Sa f(x)dx = ΣyiAx¡. If ƒ € R[a, b], show that for f(x)dx = lime→a+ √ f(x)dx.

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To show that if f in Ra b then f in Rc d for all closed subintervals c d of a b we can use the defin... View the full answer