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Semester 1: Real Analysis and Linear Algebra
Metric Space open, closed sets Intervals rectangles
Metric Space, Open and Closed Sets, Intervals, Rectangles
A metric space is a set equipped with a distance function that defines the distance between any two points in the set.
Non-negativity: Distance is always non-negative.
Identity: The distance between two points is zero if and only if the points are the same.
Symmetry: The distance from point A to point B is the same as from B to A.
Triangle inequality: The distance from A to C is less than or equal to the distance from A to B plus the distance from B to C.
Euclidean space: R^n with the standard distance metric.
Discrete metric space: A set with distance defined as 0 if points are the same, and 1 otherwise.
An open set in a metric space is a set where, for every point within the set, there exists a radius such that the ball centered at that point is entirely contained in the set.
Any union of open sets is open.
Any finite intersection of open sets is open.
Open intervals (a, b) in R.
Open balls in R^n.
A closed set contains all its boundary points, meaning it includes its limit points.
Any intersection of closed sets is closed.
Any finite union of closed sets is closed.
Closed intervals [a, b] in R.
The set of all limit points of a sequence in a metric space.
An interval (a, b) where a and b are not included.
An interval [a, b] where a and b are included.
An interval [a, b) or (a, b] where one endpoint is included and the other is not.
Any closed interval is a closed set.
Any open interval is an open set.
A rectangle in R^n can be defined as a Cartesian product of closed intervals.
Rectangles are closed sets in the standard Euclidean topology.
Open rectangles can be defined as the Cartesian product of open intervals.
In R^2, a rectangle can be defined as [a1, b1] x [a2, b2].
Open rectangles are of the form (a1, b1) x (a2, b2).
Real valued Continuous functions- Discontinuities - compact sets
Real valued Continuous functions - Discontinuities - Compact sets
Definition of Continuous Functions
A function f is continuous at a point c if for every epsilon greater than 0, there exists a delta greater than 0 such that whenever the distance between x and c is less than delta, the distance between f(x) and f(c) is less than epsilon.
Types of Discontinuities
1. Removable Discontinuity: The limit of f(x) as x approaches c exists, but f(c) is not equal to this limit. 2. Jump Discontinuity: The left-hand limit and right-hand limit at c exist, but they are not equal. 3. Infinite Discontinuity: The function approaches infinity as x approaches c.
Compact Sets
A set K in a metric space is compact if every open cover of K has a finite subcover. This is equivalent to saying that K is closed and bounded, according to the Heine-Borel theorem in Euclidean spaces.
Continuous Functions on Compact Sets
If f is a continuous function on a compact set K, then f is uniformly continuous on K. Furthermore, f attains its maximum and minimum values on K.
Importance of Discontinuities in Analysis
Discontinuities play a crucial role in understanding the behavior of functions. They affect integrability, differentiability, and the application of theorems such as the Intermediate Value Theorem.
Examples of Continuous and Discontinuous Functions
1. Continuous Function: f(x) = x^2 is continuous everywhere. 2. Discontinuous Function: f(x) = 1/x for x!=0 and f(0) = 0 is discontinuous at x = 0.
Bolzano Weirstrass theorem, Heine Borel theorem
Bolzano Weierstrass Theorem and Heine Borel Theorem
Bolzano Weierstrass Theorem
The Bolzano Weierstrass theorem states that every bounded sequence in R^n has a convergent subsequence. This theorem is essential in real analysis as it guarantees the presence of limit points and underpins many fundamental concepts such as compactness and closure of sets.
Boundedness
A set is considered bounded if there exists a real number M such that the distance between any two points in the set is less than M. This condition is crucial for the application of the Bolzano Weierstrass theorem.
Convergent Subsequences
A subsequence of a sequence is derived by selecting certain elements while preserving the original order. A sequence is convergent if its terms approach a specific value as the index increases.
Heine Borel Theorem
The Heine Borel theorem characterizes compact subsets of R^n. It states that a subset of R^n is compact if and only if it is closed and bounded. This theorem is fundamental for analyzing continuity and convergence in real analysis.
Closed Sets
A set is closed if it contains all its limit points. Closed sets are crucial in determining compactness and are often utilized in limits and continuity.
Relation Between Both Theorems
Both theorems emphasize important properties of sets in R^n, focusing on boundedness and convergence. The Bolzano Weierstrass theorem applies to sequences, while the Heine Borel theorem provides a criterion for compactness in terms of closure and boundedness.
Derivatives, maxima and minima, Riemann integral, Riemann Stieltjes integral
Derivatives, Maxima and Minima, Riemann Integral, Riemann-Stieltjes Integral
Derivatives
Maxima and Minima
Riemann Integral
Riemann-Stieltjes Integral
