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Problem 1

# Let $$\mathcal{X}$$ be an open subset of $$\mathbb{K}^{n}$$ and let $$\mathcal{U}$$ be a metric space. $$A$$ right-hand side (rhs) with respect to $$X$$ and $$\mathcal{U}$$ is a function $$f: \mathbb{R} \times \mathcal{X} \times \mathcal{U} \rightarrow \mathbb{K}^{n}$$ which can be obtained in the following way: There must exist another metric space $$\mathcal{S}$$ as well as maps $$\tilde{f}: \mathcal{S} \times x \times \mathcal{U} \rightarrow \mathbb{K}^{n}$$ and $$\pi: \mathbb{R} \rightarrow \mathcal{S}$$ so that $$f(t, x, u)=\tilde{f}(\pi(t), x, u)$$ and the following properties hold: 1\. $$\widetilde{f}(s, \cdot, u)$$ is of class $$\mathcal{C}^{1}$$ for each fixed $$s, u$$; 2\. both $$\tilde{f}$$ and the partial derivative $$\tilde{f}_{x}$$ are continuous on $$\mathcal{S} \times \mathcal{X} \times \mathcal{U}$$; 3\. $$\pi$$ is a measurable locally essentially bounded function.

Expert verified
To verify that the given function $$f: \mathbb{R} \times \mathcal{X} \times \mathcal{U} \rightarrow \mathbb{K}^{n}$$ is an rhs with respect to $$X$$ and $$\mathcal{U}$$, we must check the following properties: 1. For each fixed $$s,u$$, $$\tilde{f}(s, \cdot, u)$$ is of class $$\mathcal{C}^{1}$$, meaning it is differentiable and its derivative is continuous. 2. Both $$\tilde{f}$$ and the partial derivative $$\tilde{f}_{x}$$ are continuous on $$\mathcal{S} \times \mathcal{X} \times \mathcal{U}$$. 3. $$\pi$$ is a measurable locally essentially bounded function. If these properties hold, then $$f$$ is indeed an rhs with respect to $$X$$ and $$\mathcal{U}$$.
See the step by step solution

## Step 1: Verify $$\tilde{f}(s, \cdot, u)$$ is of class $$\mathcal{C}^{1}$$ for each fixed $$s, u$$.

In order to verify this, we need to ensure that for each fixed $$s$$ and $$u$$, $$\tilde{f}(s, \cdot, u)$$ is differentiable and its derivative is continuous. This means the function behaves "smoothly", without abrupt changes or jumps when we fix $$s$$ and $$u$$. The requirement for $$\tilde{f}(s, \cdot, u)$$ to be of class $$C^1$$ separates right-hand side (RHS) functions from ordinary functions, as RHS functions have to be "smooth" with respect to the space variable $$x$$.

## Step 2: Verify $$\tilde{f}$$ and the partial derivative $$\tilde{f}_{x}$$ are continuous on $$\mathcal{S} \times \mathcal{X} \times \mathcal{U}$$.

This requirement ensures that both the function $$\tilde{f}$$ and its partial derivatives change smoothly and gradually throughout the entire space $$\mathcal{S} \times \mathcal{X} \times \mathcal{U}$$. If they weren't, $$\tilde{f}$$ would not be a well-behaved rhs with respect to $$X$$ and $$\mathcal{U}$$. To check this, we need to evaluate the limit of $$\tilde{f}$$ and $$\tilde{f}_x$$ in the neighborhood of any point in $$\mathcal{S} \times \mathcal{X} \times \mathcal{U}$$ and confirm it equals the value of $$\tilde{f}$$ or $$\tilde{f}_x$$ at that point.

## Step 3: Verify $$\pi$$ is a measurable locally essentially bounded function.

This means that for any set of measurement (i.e. a set where we can reasonably talk about its size or volume), the function $$\pi$$ is essentially bounded (its values don't go to infinity). You can check this by picking any subset of the real numbers $$\mathbb{R}$$ and confirming that $$\pi$$ doesn't take on any infinite values on this subset (except possibly on a set of measure zero - 'essentially' bounded means we're allowed to ignore sets of measure zero). This property ensures that the mappings from $$\mathbb{R}$$ to $$\mathcal{S}$$, which give us the structure of the rhs $$f$$, are reasonable and not prone to sudden, drastic changes. If the functions $$\pi$$ and $$\tilde{f}$$ meet all these criteria, then $$f$$ is indeed an rhs with respect to $$X$$ and $$\mathcal{U}$$. If not, it does not qualify as an rhs.

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