1. Let \begin{equation*} f(x, y) = \begin{cases} \frac{y \sin x}{x} & (x, y) \neq (0, y) \\ y & (x, y) = (0, y) \end{cases} \end{equation*} a. Is $f$ is continuous for all $(x, y) \in \mathbb{R}$? b. Calculate $f_x$ and $f_y$. Does $f_x, f_y$ exists for all $(x, y) \in \mathbb{R}$?
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To determine if f(x,y) is continuous for all (x,y) ∈ R, we need to check if the limit of f(x,y) as (x,y) approaches any point (a,b) exists and is equal to f(a,b). Let's consider a point (a,b) ∈ R. We want to find the limit of f(x,y) as (x,y) approaches Show more…
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