Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be given by \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{n}}\) where $\mathrm{n} \in \mathrm{N}$, the set of natural numbers. Prove that \(\mathrm{f}\) is everywhere continuous.

Let \(\mathrm{F}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+3 \mathrm{y}\) be defined on the unit square $$ \mathrm{S}=\\{(\mathrm{x}, \mathrm{y}): 0 \leq \mathrm{x} \leq 1,0 \leq \mathrm{y} \leq 1\\} $$ Show that \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) is continuous on \(\mathrm{S}\).

Show that the exponential function defined by $$ \exp \mathrm{x}=\lim _{\mathrm{n} \rightarrow \infty}[1+(\mathrm{x} / \mathrm{n})]^{\mathrm{n}} $$ is everywhere continuous.

Let \(\mathrm{f}\) be the real-valued function defined on \([0,1]\) given by \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{2} \sin (1 / \mathrm{x})\) for \(\left.\mathrm{x} \in(0,1)\right]\) and \(\mathrm{f}(0)=0\). Show that \(\mathrm{f}\) is absolutely continuous on \([0,1]\).

Which of the following functions are uniformly continuous on the specified domains? a) \(f(x)=x^{3}(0 \leq x \leq 1)\) b) \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}(0 \leq \mathrm{x}<\infty)\) c) \(f(x)=\sin x^{2}(0 \leq x<\infty)\) d) \(f(x)=1 /\left(1+x^{2}\right)(0 \leq x<\infty)\)

Consider the function \(\mathrm{f}:[0,1] \rightarrow[0,1]\) given by $$ \begin{aligned} &\mathrm{f}(\mathrm{x})=1 / \mathrm{q}, \mathrm{q}>0, \mathrm{x}=\mathrm{p} / \mathrm{q}, \text { i.e., rational } \\ &\mathrm{f}(\mathrm{x})=0 \\ &\mathrm{x} \text { irrational, } \mathrm{x} \in \mathrm{R}-\mathrm{Q} \end{aligned} $$ a) Show that \(\mathrm{f}\) is discontinuous at any rational number. b) Show that \(\mathrm{f}\) is continuous at each irrational.