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Relation between primitivableness and integrableness for certain functions

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October 28, 2019
functionprimitivablenessantiderivativedifferentiabilityIntegralIndefinite integralreal analysis

Problem Statement

Let be two real numbers a<b a<b and a function f:[a,b]⟶R f:[a,b]\longrightarrow\mathbb{R} having the property that if the sequence (f(xn))n≥1 \left(f\left( x_n \right)\right)_{n\ge 1} is convergent, then the sequence (xn)n≥1 \left( x_n \right)_{n\ge 1} is convergent.
a) Prove that if f f admits antiderivatives, then f f is integrable. b) Is the converse of a) true?
Marcelina Popa
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