• Title: Uniform Convergence for Differentiable Functions

  • Series: Real Analysis

  • Chapter: Differentiable Functions

  • YouTube-Title: Real Analysis 37 | Uniform Convergence for Differentiable Functions

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

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  • Quiz Content

    Q1: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of functions $f_n: [0,1] \rightarrow \mathbb{R}$. Which implication is correct?

    A1: If it is pointwisely convergent, then also uniformly convergent.

    A2: If it is uniformly convergent, then also pointwisely convergent.

    Q2: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of continuous functions $f_n: [0,1] \rightarrow \mathbb{R}$. Which implication is correct?

    A1: If it is pointwisely convergent to $f$, then $f$ is also continuous.

    A2: If it is uniformly convergent to $f$, then $f$ is also continuous.

    A3: If it is uniformly convergent to $f$, then $f$ is also differentiable.

    Q3: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of differentiable functions $f_n: [0,1] \rightarrow \mathbb{R}$. If the sequence is uniformly convergent to $f: [0,1] \rightarrow \mathbb{R}$, is $f$ also differentiable?

    A1: No, never!

    A2: Yes, always!

    A3: In general, it is not.

    Q4: Let $(f_1, f_2, f_3, \ldots)$ be a sequence of differentiable functions $f_n: [0,1] \rightarrow \mathbb{R}$ that is pointwisely convergent to $f: [0,1] \rightarrow \mathbb{R}$. Which implication is correct?

    A1: If $(f_1^\prime, f_2^\prime, f_3^\prime, \ldots)$ is uniformly convergent, then $f$ is also differentiable.

    A2: If $(f_1^\prime, f_2^\prime, f_3^\prime, \ldots)$ is pointwisely convergent, then $f$ is also differentiable.

    A3: If $(f_1^\prime, f_2^\prime, f_3^\prime, \ldots)$ is pointwisely convergent, then $f$ is not differentiable.

  • Last update: 2024-10

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