Let \(f: R^{2} \rightarrow R\) be given by $$ f(x, y)=x^{2}+y^{2} $$ Show that \(\mathrm{f}\) is continuous at \((0,0)\).

Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+3 \mathrm{y}\) and let \(\varepsilon>0\) be given. Prove that \(\mathrm{F}\) is continuous in the whole plane by finding \(\delta>0\) such that for $\left|(x, y)-\left(x_{0}, y_{0}\right)\right|<\delta$ \(\left|F(x, y)-F\left(x_{0}, y_{0}\right)\right|<\varepsilon\) where $x_{0}, y_{0}$ is an arbitrary point in the plane.

Let \(\mathrm{I}=[0,1]\), the closed unit interval on the real line. Then \(\mathrm{I} \times \mathrm{I}=\mathrm{I}^{2}\) is the unit square in \(\mathrm{R}^{2}\). Show that there exists a continuous function $\mathrm{f}: \mathrm{I} \rightarrow \mathrm{I}$ which is subjective, i.e. the image of \(\mathrm{f}\) is the square.

Show that the function \(\mathrm{f}(\mathrm{x})=1 / \mathrm{x}\) is not uniformly continuous on the half-open interval \((0,1]\) but is uniformly continuous on \([1, b]\) where \(b \in R\). What is a sufficient condition for functions defined on subsets of \(\mathrm{R}^{\mathrm{n}}\) to be uniformly continuous?

Show that $$ \lim _{(x, y) \rightarrow(0,0)}\left[\left(2 x^{3}-y^{3}\right) /\left(x^{2}+y^{2}\right)=0\right. $$

Show that a continuous function of a continuous function is continuous by means of the following examples (i.e., show that g \(^{\circ} \mathrm{f}\) is continuous): (a) \(f(x, y)=(1+x y)^{2} g(z)=\sin z\) (b) \(\mathrm{f}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{3}\) is defined by $$ \begin{aligned} &\mathrm{f}(\mathrm{x}, \mathrm{y})=\left(\mathrm{x}, \mathrm{y}, \mathrm{x}\left(1+\mathrm{x}^{2}+\mathrm{y}^{2}\right)-3 / 2\right) \\ &\mathrm{g}: \mathrm{R}^{3} \rightarrow \mathrm{R} \text { defined by } \\ &\mathrm{g}(\mathrm{x}, \mathrm{y}, z)=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2} \end{aligned} $$