Limits and Continuity

Explore the fundamental concepts of limits and continuity in calculus through a comprehensive 9-hour course. Delve into epsilon-delta definitions, various types of continuity, and key theorems. Master techniques for proving continuity, analyze challenging examples, and understand advanced topics like uniform continuity and the Darboux Integral. Apply these concepts to solve complex problems and gain a deep understanding of the mathematical foundations crucial for advanced calculus and analysis.

Epsilon Delta Continuity (Example 1).
The Ultimate Continuity Battle (Example 2).
f is not continuous (Example 3).
Epsilon Delta Continuity (Example 4): x^3.
Epsilon Delta Continuity (Example 5): Absolute Value.
Epsilon Delta Continuity (Example 6): 1/x.
Epsilon Delta Continuity (Example 7): Square Root.
Sequential vs Epsilon-Delta Continuity.
f+g is continuous.
fg is continuous.
f/g is continuous.
Chen Lu is continuous.
Max is continuous.
The Popcorn Function.
f(2x) = f(x).
f(x^2) = f(x).
Continuous implies Bounded.
Can you prove it? The Extreme Value Theorem.
Can you prove it? The Intermediate Value Theorem.
Image of an interval.
What is a fixed point?.
Calculus Square Root.
a pyramid fact about functions.
Continuity and Monotonicity.
f inverse is continuous.
The Borsuk Ulam Theorem.
What is Uniform Continuity?.
Uniform Continuity (Example 1): The Basics.
Uniform Continuity (Example 2): Challenging Example.
Uniform Continuity (Example 3): Not Uniformly Continuous.
Uniform Continuity (Example 4): Not Uniformly Continuous.
Hölder Continuity.
Uniform Continuity and Compactness.
Voulez-vous Cauchy avec moi?.
What is a continuous extension?.
a neat fact about uniform continuity.
The Darboux Integral.
Epsilon delta limit (Example 1).
Epsilon delta limit (Example 2).
Epsilon delta limit (Example 3): Infinite limit at a point.
Epsilon delta limit (Example 4): Limits at infinity.
Epsilon delta limit (Example 5): x cubed.
Epsilon delta limit (Example 6): Square Roots.
Epsilon delta limit (Example 7): 1/x.
How do you prove it? The Squeeze Theorem.