Choosing parameters for continuity of a piecewise-defined function

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Find the value of $a$ for which the function

u = \begin{cases} \frac{1}{x+y} e^{-1 /\left|x+y\right|}, & \text{ if } & x+y \neq 0 \\ a, & \text{ if } & x+y = 0 \end{cases}

is continuous in $R^{2}$ .

Find the values of $a$ and $b$ for which the function

u = \begin{cases} a, & \text{ if } x^{2}+y^{2} \leqslant 4, \\ \sqrt{9-x^{2}-y^{2}}-\sqrt{x^{2}+y^{2}-4}, & \text{ if } 4 < x^{2}+y^{2} \leqslant 9, \\ b, & \text{ if } x^{2}+y^{2} > 9, \end{cases}

is continuous in $R^{2}$ .

Find the values of $a$ and $b$ for which the function

u = \begin{cases} a, & \text{ if } x^{2}+y^{2} = 0, \\ \sqrt{5\left(x^{2}+y^{2}\right)-4-\left(x^{2}+y^{2}\right)^{2}}, & \text{ if } 1 \leqslant x^{2}+y^{2} < 4, \\ b, & \text{ if } x^{2}+y^{2} \geqslant 4, \end{cases}

is continuous on its domain of definition.

Find the values of $a$ and $b$ for which the function

u = \begin{cases} a, & \text{ if } & \sum _{i = 1}^{n} x_{i}^{2} = 0, \\ \sum _{i = 1}^{n} x_{i}^{2} / \ln \left(1-\sum _{i = 1}^{n} x_{i}^{2}\right), & \text{ if } & 0 < \sum _{i = 1}^{n} x_{i}^{2} < 1, \\ b, & \text{ if } & \sum _{i = 1}^{n} x_{i}^{2} \geqslant 1, \end{cases}

is continuous in $R^{n}$ .

References

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.