Question

Let $F: \mathbb R^m \to \mathbb R^n$ be smooth, $F(0) = 0$. Let $H(t,x) = \frac{F(tx)}{t}$ if $t \neq 0$ and $DF|_0(x)$ otherwise. Show $H$ is smooth

score 50 · Answer 1 · 2026-04-15 18:19:53

50

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Let $F: \mathbb R^m \to \mathbb R^n$ be smooth, $F(0) = 0$. Let $H(t,x) = \frac{F(tx)}{t}$ if $t \neq 0$ and $DF|_0(x)$ otherwise. Show $H$ is smooth

Published on 15 Apr 2026 - 18:19

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113

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87

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17

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41

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112

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List Question

Let $F: \mathbb R^m \to \mathbb R^n$ be smooth, $F(0) = 0$. Let $H(t,x) = \frac{F(tx)}{t}$ if $t \neq 0$ and $DF|_0(x)$ otherwise. Show $H$ is smooth

Solving a functional equation $\alpha f(x + y) = f(x)f(y)$

Did I perform the partial derivative correctly?

Expressing the gradient of the softmax cost function using matrix multiplication

Partial derivative of $f(x,y(x))$

Question about function of two variables.

What's the Derivative of a Vector in a function with summation squared

Help finding second directional derivative

Partial Derivative of a Product of a Sequence

Kurzweil-Hensotck integral, an unclear point in a book for undergraduates by Fonda

Vanishing second derivative at a point, but non-vanishing first derivative?

Why is it possible to cancel a function in $\frac{\partial f}{\partial x}$?

Solving for $u$ and $v$ in the 2D Jacobian

Partial derivative of endogenous variables and total differentiation

How to partially differentiate an equation with sums?

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