-
Title: Continuity
-
Series: Multivariable Calculus
-
YouTube-Title: Multivariable Calculus 2 | Continuity
-
Bright video: https://youtu.be/1_A3UBuyFhY
-
Dark video: https://youtu.be/0j1CxaJSJjQ
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: mc02_sub_eng.srt missing
-
Timestamps
0:00 Intro
0:25 Continuous Functions
1:45 Continuity via sequences
2:39 Measuring distance in ℝⁿ
5:31 Convergent sequences in ℝⁿ
8:20 (Non-trivial) Link between single-variable convergence definition vs. new definition
10:33 Multivariable continuity
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Which of the following statements is equivalent to the statement: $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous at $x_0$.
A1: For all convergent sequences $(x_n) \subseteq \mathbb{R}$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} f(x_n) = f(x_0)$.
A2: For all convergent sequences $(x_n) \subseteq \mathbb{R}\setminus{ x_0 }$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} f(x_n) = x_0$.
A3: For all convergent sequences $(x_n) \subseteq \mathbb{R}\setminus{ x_0 }$ with limit $x_0$, we have $\lim_{n \rightarrow \infty} x_n = x_0$.
A4: For all convergent sequences $(f(x_n)) \subseteq \mathbb{R}\setminus{ x_0 }$ with limit $f(x_0)$, we have $\lim_{n \rightarrow \infty} x_n = x_0$.
Q2: What is the correct definition of the Euclidean distance $d_{\text{ Euclid}}$ in $\mathbb{R}^n$?
A1: $$d_{\text{ Euclid}}(\mathbf{x}, \mathbf{y}) = \sqrt{\sum_{j = 1}^n (x_j - y_j)^2}$$
A2: $$ d_{\text{ Euclid}}(\mathbf{x}, \mathbf{y}) = \mathbf{x} - \mathbf{y} $$
A3: $$ d_{\text{ Euclid}}(\mathbf{x}, \mathbf{y}) = 0 $$
A4: $$ d_{\text{ Euclid}} = \sum_{j = 1}^n (x_j - y_j)^2 $$
Q3: Which of the following statements is the definition that $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $\mathbf{x} \in \mathbb{R}^n$?
A1: For all convergent sequences $(\mathbf{x}^{(k)}) \subseteq \mathbb{R}^n$ with limit $x$, we have $\lim_{k \rightarrow \infty} f(\mathbf{x}^{(k)}) = \mathbf{x}$.
A2: For all convergent sequences $(\mathbf{x}^{(k)}) \subseteq \mathbb{R}^n$ with limit $x$, we have $\lim_{k \rightarrow \infty} \mathbf{x}^{(k)} = f(\mathbf{x})$.
A3: For all convergent sequences $(\mathbf{x}^{(k)}) \subseteq \mathbb{R}^n$ with limit $x$, we have $\lim_{k \rightarrow \infty} f(\mathbf{x}^{(k)}) = f(\mathbf{x})$.
-
Last update: 2024-10