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Let $f$ be a two-variable real-valued function on disk $D \subset \mathbb{R^2}$ and $(a,b)$ is the center of $D$.

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The first problem is about continuity and partial differentiability. At $(a,b)$, we have 3 concerned "objects" of $f$: (respectively) continuity, $x$-partial differentiability, $y$-partial differentiability. Each of them has 2 states: $1$ for "able" and $0$ for "unable". In my sense, a continuously-elementary $f$ has the ($1$,$1$,$1$) state of those objects. Therefore:

Problem 1: Find an example (if any) of each of the non-continuously-elementary types of $f$. If there's no example of one of the types, prove that case.

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Now assume that $f$ has 2 partial derivatives on $D$. The second problem is about differentiability and continuity of the partial derivatives. At $(a,b)$, now we have 3 concerned "objects" of $f$: (respectively) differentiability, continuity of the $x$-partial derivative, continuity of the $y$ partial derivative. Also in my sense, a differentiably-elementary $f$ has the ($1$,$1$,$1$) state of those objects. Therefore:

Problem 2: Find an example (if any) of each of the non-differentiably-elementary types of $f$. If there's no example of one of the types, prove that case.

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Note:

$i)$ I came up with these problems while studying, so it's not an assignment. Also, the types are symbolized well enough, you can use those symbols.

$ii)$ In Problem 1, because of the symmetry of two variables, there are $6-1 = 5$ types of non-elementary $f$ instead of $2^3-1 = 7$.

$iii)$ With the same argument, in Problem 2, there are $5$ types of non-elementary $f$. However, the type of ($0$,$1$,$1$) can be omitted by the familiar lemma. Hence, $4$ types are left.

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Thanks in advance!

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