• Title: Lipschitz Continuity

  • Series: Ordinary Differential Equations

  • YouTube-Title: Ordinary Differential Equations 9 | Lipschitz Continuity

  • Bright video: https://youtu.be/d1dvGoqyXLM

  • Dark video: https://youtu.be/tN1BKhqrmLE

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ode09_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $v: \mathbb{R} \rightarrow \mathbb{R}$ be a locally Lipschitz continuous function. What is not correct in general?

    A1: $v$ is continuously differentiable.

    A2: $v$ is continuous.

    A3: For every point $x \in \mathbb{R}$ there is an $\varepsilon > 0$ and a constant $L > 0$ such that for all points $y,z \in (x- \varepsilon, x+\varepsilon)$, we have $$ |v(y) - v(z) | \leq L | y - z | $$

    Q2: Let $v: \mathbb{R} \rightarrow \mathbb{R}$ be given by $v(x) = x^2$. Is $v$ locally Lipschitz continuous?

    A1: Yes!

    A2: No!

    A3: One needs more information.

    Q3: Let $v: \mathbb{R} \rightarrow \mathbb{R}$ be given by $v(x) = \sqrt{|x|}$. Is $v$ locally Lipschitz continuous?

    A1: No!

    A2: Yes

    A3: One needs more information.

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