• Title: Examples of Continuous Functions

  • Series: Multivariable Calculus

  • YouTube-Title: Multivariable Calculus 3 | Examples of Continuous Functions

  • Bright video: https://youtu.be/KkZFMklbiu0

  • Dark video: https://youtu.be/xopJmjXvicc

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: mc03_sub_eng.srt missing

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  • Quiz Content

    Q1: Is a function $f: \mathbb{Z} \rightarrow \mathbb{R}^2$ continuous?

    A1: Yes, always!

    A2: No, never!

    A3: There are continuous functions of this form but also functions that are not continuous.

    A4: The notion ‘continuous’ does not make sense for such functions.

    Q2: Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$ f(x_1, x_2) = x_1 \cdot x_2$$ continuous?

    A1: Yes!

    A2: No!

    A3: One needs more information.

    Q3: Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$ f(x_1, x_2) = \begin{cases} 1 ~~ \text{ if } \binom{x_1}{x_2} = \binom{0}{0} \ 2 ~~ \text{ if } \binom{x_1}{x_2} \neq \binom{0}{0} \end{cases} $$ continuous?

    A1: Yes!

    A2: No!

    A3: One needs more information.

    Q4: Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $$ f(x_1, x_2) = \begin{cases} 0 ~~ \text{ if } \binom{x_1}{x_2} = \binom{0}{0} \ \frac{x_1 x_2}{x_1^2 + x^2_2} ~~ \text{ if } \binom{x_1}{x_2} \neq \binom{0}{0} \end{cases} $$ continuous?

    A1: Yes!

    A2: No!

    A3: One needs more information.

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