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Category Archives: 01 Real Analysis
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 and … Continue reading
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 theorem as A notation that … Continue reading
Posted in 01 Real Analysis, 02 Mathematics
Tagged analysis, math, mathematics, Physics
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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 interpretation of the … Continue reading
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 speaking the previous theorem states that the graph … Continue reading
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 … Continue reading
Real Analysis – Limits and Continuity V
The condition is somewhat hard to get into our heads as neophytes. On top of that the similarity of the definition for limit and continuity can increase the confusion and to try to counter those frequent turn of events the … Continue reading
Posted in 01 Real Analysis, 02 Mathematics
Tagged inequalities, math, mathematics, Physics
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Real Analysis Exercises III
1. a) Calculate and As we can see the first term cancels out with the fourth, the third with the sixth, and so on and all we are left with is the second and second last terms: b) Calculate using … Continue reading
Real Analysis – Limits and Continuity IV
As an application of theorem 35 let us look into the functions and . Now and is a strictly increasing function, and also is a strictly increasing function. By theorem 35 it is and . As for it is and … Continue reading
Real Analysis – Limits and Continuity III
The first thing I want to say is that the concept of limit is a local concept. In mathematical lingo what this means is that for a function to have a limit in a given point, , it doesn’t matter … Continue reading
Real Analysis – Limits and Continuity II
In this case it is and . If is a sequence of points in such as it follows . After this simple example we’ll introduce a theorem that will state a somewhat obvious result. In layman terms what it expresses … Continue reading
From Suits to Spectra 