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Real Analysis – Limits and Continuity VI
— More properties of continuous functions — Definition 35 Let ; and . If , we can define as: As an application of the previous definition let us look into . It is . Since we can define as As another...
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Real Analysis Limits and Continuity VII
— 6.10. Global properties of continuous functions — Theorem 51 (Intermediate Value Theorem) Let and is a continuous function. Let such that , then there exists such that .Proof: Omitted. Intuitively...
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Real Analysis – Differential Calculus I
— 7. Differential Calculus — Definition 36 Let , and . is differentiable in point if the following limit exists This limit is represented by and is said to be the derivative of in . The geometric...
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Real Analysis – Differential Calculus II
Theorem 60 (Differentiability of the composite function) Let , , and . If is differentiable in and is differentiable in , then in and it is Using Leibniz’s notation we can also write the previous...
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Real Analysis – Differential Calculus III
Theorem 65 (Cauchy’s theorem) Let and , continuous such as . If and are differentiable in and doesn’t vanish in , there exists such as Proof: It is since if it were would vanish in by Theorem 63. Let...
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