• 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

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

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  • 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

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