Chapter 3

Find a) \(\lim _{\mathrm{x} \rightarrow \mathrm{C}}\left[\left(\mathrm{x}_{\mathrm{n}}-\mathrm{C}_{\mathrm{n}}\right) /(\mathrm{x}-\mathrm{C})\right]\) b) \(\lim _{\mathrm{x} \rightarrow 0}\left[\left(\mathrm{x}^{2}\right) /\left\\{\left(\mathrm{x}^{2}+1\right)^{(1 / 2)}-1\right\\}\right] .\)

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

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

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 \(\mathrm{F}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+3 \mathrm{y}\) 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 \(\mathrm{S}\).

Let \(\mathrm{f}: \mathrm{R}^{2} \rightarrow \mathrm{R}\) be a real valued function with domain \(\mathrm{R}^{2}\) which is of the form $$ f\left(x_{1}, x_{2}\right)=\left[\left(5 x_{1}\right) /\left(1+x^{2} 2\right)\right] $$ Show that \(\mathrm{f}\) is continuous at \((0,0)\).

Let \(\mathrm{f}: \mathrm{R}^{3} \rightarrow \mathrm{R}\) be given by \(\mathrm{f}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left[\left(\mathrm{y}^{3} \mathrm{z}\right) /\left(1+\mathrm{x}^{2}+\mathrm{z}^{3}\right)\right]\) ] Show that \(\mathrm{f}\) is continuous at \((0,0,0)\).

Show that the function \(\mathrm{f}: \mathrm{R}^{n} \rightarrow \mathrm{R}\) given by \(\mathrm{f}\left(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\right)=\max \left\\{\mathrm{x}_{1}, \ldots ., \mathrm{x}_{\mathrm{n}}\right\\}\) is continuous everywhere.

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.

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.

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 $$ \begin{aligned} &\mathrm{f}(\mathrm{x}, \mathrm{y})=\left(\mathrm{x}, \mathrm{y}, \mathrm{x}\left(1+\mathrm{x}^{2}+\mathrm{y}^{2}\right)-3 / 2\right) \\ &\mathrm{g}: \mathrm{R}^{3} \rightarrow \mathrm{R} \text { defined by } \\ &\mathrm{g}(\mathrm{x}, \mathrm{y}, z)=\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2} \end{aligned} $$

Let \(\mathrm{f}(\mathrm{x}, \mathrm{y})\) be given by $$ \begin{array}{cl} \mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy}^{2} /\left(\mathrm{x}^{2}+\mathrm{y}^{4}\right) & (\mathrm{x}, \mathrm{y}) \neq(0,0) \\ \text { and }=0 & \mathrm{x}=\mathrm{y}=0 \end{array} $$ Is f continuous at the origin?

Distinguish between removable and essential discontinuities. Decide, for the following functions whether the points of discontinuity are a) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy} / \mathrm{x}^{2}+\mathrm{y}^{2} \quad(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2}-\\{(0,0)\\}\) b) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2} \mathrm{y}^{2} / \mathrm{x}^{2}+\mathrm{y} \quad(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2}-\\{(0,0)\\}\) c) \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}}\) \(x \in R, x>0\) d) \(g(x) \quad=x\) \(\mathrm{x} \in \mathrm{R}, \mathrm{x}>0\) and \(=2 \quad x \in R, x=0\) e) \(h(x)=\sin (1 / x) \quad x \in R, x<0\)

What is meant by a discontinuity of the first kind? A discontinuity of the second kind? Show that monotonic functions have no discontinuities of the second kind.

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.

The Baire category theorem states that the real line is of the second category. Show how the Baire category theorem can be used to prove the following theorem: There exists no \(\mathrm{F}: \mathrm{R} \rightarrow \mathrm{R}\) such that \(\mathrm{f}\) is continuous at each rational but discontinuous at each irrational.

The Baire category theorem states that the real line is of the second category. Show how the Baire category theorem can be used to prove the following theorem: There exists no \(\mathrm{F}: \mathrm{R} \rightarrow \mathrm{R}\) such that \(\mathrm{f}\) is continuous at each rational but discontinuous at each irrational.

To which class of continuous functions do the following functions belong? a) Weierstrass' s nowhere differentiable function b) \(\mathrm{f}(\mathrm{x}, \mathrm{y}, z)=\mathrm{e}^{\mathrm{x}-\mathrm{y}+\mathrm{z}}\) c) \(f(t)=t^{3} e_{1}+(\sin t) e_{2}+t^{8 / 3} e_{3},-\infty

Prove the Intermediate Value Theorem for the derivative of a real differentiable function on \([a, b]\).

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?

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be given by \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}\) Show that although \(\mathrm{f}\) is continuous at every point in the domain it is not uniformly continuous.

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) \(f(x)=1 /\left(1+x^{2}\right)(0 \leq x<\infty)\)

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 \([0,1]\).

The Arezela-Ascoli theorem states that a set \(E\) in \(C(X, R)\) is compact if and only if it is closed, bounded and equicontinuous. Let \((\mathrm{X}, \mathrm{d})\) be the metric space consisting of all continuous functions from the compact metric space \(\mathrm{X}\) into \(\mathrm{R}^{\mathrm{n}}\) with distance function given by \(\mathrm{d}(\mathrm{f}, \mathrm{g})=\sup \\{\mathrm{I}|\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x})| \mid: \mathrm{x} \in \mathrm{X}\\}\). State the Arezela-Ascoli theorem for this space.

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 why the function, \(\mathrm{f}(\mathrm{x})={ }^{\infty} \Sigma_{\mathrm{n}=0} \mathrm{a}^{\mathrm{n}} \cos \left(b^{\mathrm{n}} \pi \mathrm{x}\right) \quad 0<\mathrm{a}<1 \quad \mathrm{~b}=Z \mathrm{k}-1\) constructed by Weierstrass, is continuous everywhere but differentiable nowhere.

Construct an everywhere continuous function that is nowhere differentiable from the following functions \(\mathrm{K}(\mathrm{x})\) is the distance from \(\mathrm{x}\) to the nearest integer, \(\mathrm{K}: \mathrm{R} \rightarrow \mathrm{R}\).