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Sahithyan's S1
Sahithyan's S1 — Mathematics

Riemann Integrable

A bounded function f:[a,b]Rf:[a,b]\rightarrow\mathbb{R} is Riemann integrable on [a,b][a,b] iff U(f)=L(f)U(f)=L(f). In that case, the Riemann integral of ff on [a,b][a,b] is denoted by:

abf(x)dx\int_{a}^{b}{f(x)\,\text{d}x}

Reimann Integrable or not

FunctionYes or No?How?
UnboundedNoBy definition
ConstantYesP(any partition)    L(f;P)=U(f;P)\forall P\,\text{(any partition)}\;\; L(f;P)=U(f;P)
Monotonically increasing/decreasingYesTake a partition such that Δx<δ=ϵf(b)f(a)\Delta{x}<\delta=\frac{\epsilon}{f(b)-f(a)}
ContinuousYesTake a partition such that Δx<δ=ϵ2(ba)\Delta{x}<\delta=\frac{\epsilon}{2(b-a)}