-
Recent Posts
Archives
- March 2023
- January 2023
- May 2020
- April 2020
- March 2020
- June 2018
- July 2016
- June 2016
- May 2016
- March 2015
- February 2015
- January 2015
- December 2014
- December 2013
- November 2013
- July 2013
- June 2013
- May 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
Categories
Meta
Pages
Tag Archives: continuity
Commutative Algebra 5
Morphisms in Algebraic Geometry Next we study the “nice” functions between closed subspaces of . Definition. Suppose and are closed subsets. A morphism is a function which can be expressed as: for some polynomials . We also say f is a regular … Continue reading
Posted in Advanced Algebra
Tagged algebraic geometry, continuity, ideals, morphisms, rings, topology
2 Comments
Topology: Hausdorff Spaces and Dense Subsets
Hausdorff Spaces Recall that we’d like a condition on a topological space X such that if a sequence converges, its limit is unique. A sufficient condition is given by the following: Definition. A topological space X is said to be Hausdorff if … Continue reading
Posted in Notes
Tagged advanced, continuity, dense subsets, Hausdorff, metric spaces, nets, topology
4 Comments
Topology: Nets and Points of Accumulation
Recall that a sequence in a topological space X converges to a in X if the function f : N* → X which takes is continuous at . Unrolling the definition, it means that for any open subset U of X containing a, the set contains (N, ∞] for some N. In … Continue reading
Posted in Notes
Tagged advanced, closed subsets, continuity, convergence, limits, metric spaces, nets, points of accumulation, sequences, topology
Leave a comment
Topology: Limits and Convergence
Following what we did for real analysis, we have the following definition of limits. Definition of Limits. Let X, Y be topological spaces and . If f : X-{a} → Y is a function, then we write if the function: is … Continue reading
Posted in Notes
Tagged advanced, continuity, convergence, extended reals, Hausdorff, limits, sequences, topology
Leave a comment
Topology: Continuous Maps
Continuity in Metric Spaces Following the case of real analysis, let’s define continuous functions via the usual ε-δ definition. Definition. Let (X, d) and (Y, d’) be two metric spaces. A function f : X → Y is said to be … Continue reading
Posted in Notes
Tagged advanced, continuity, disjoint union topology, homeomorphism, metric spaces, product topology, subspaces, topology
Leave a comment
Basic Analysis: Closed Subsets and Uniform Continuity
Let’s consider another question: suppose f : D → R is continuous, where D is a subset of R. If (xn) is a sequence in D converging to some real L, is it true that (f(xn)) is also convergent? Now if L is in D, then we know that (f(xn)) → (f(L)). … Continue reading
Posted in Notes
Tagged advanced, analysis, closed subsets, continuity, points of accumulation, uniform continuity, uniform convergence
Leave a comment
Basic Analysis: Uniform Convergence
Once again, let be a subset. Suppose we now have a sequence of functions , where n = 1, 2, 3, … , such that for each x in D, the sequence converges to some real value. We’ll denote this value by f(x), thus … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, convergence, pointwise convergence, series, uniform convergence
1 Comment
Basic Analysis: Limits and Continuity (3)
Let’s consider multivariate functions where . To that end, we need the Euclidean distance function on Rn. If x = (x1, x2, …, xn) is in Rn, we define: Note that |x| = 0 if and only if x is the zero vector 0. Now we are ready … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, limits, multivariate, open balls, open subsets, topology
Leave a comment
Basic Analysis: Limits and Continuity (2)
Previously, we defined continuous limits and proved some basic properties. Here, we’ll try to port over more results from the case of limits of sequences. Monotone Convergence Theorem. If f(x) is increasing on the open interval (c, a) and has … Continue reading
Posted in Notes
Tagged advanced, analysis, continuity, limits, monotone convergence theorem, points of accumulation, squeeze theorem
Leave a comment
Mathematics and Such 