B.Sc. Mathematics Honours C-5 Question Paper 2018 (CBCS)

Theory of Real Functions and Introduction to Metric Space

UNIT-1

1. Answer any Three questions: 3 x 2
(a) Prove that $lim_{x\rightarrow0} x sin(\frac{1}{x^{2}})=0$.
(b) Let $D\subset R$ and $f:D\rightarrow R$ be a function. If c be an isolated point of D then prove that f is continuous at c.
(c) State the sequential criterion for the continuity of a function f at a point c.
(d) By Cauchy's principle prove that $lim_{x\rightarrow0}cos\frac{1}{x}$ does not exist.
(e) Show that $f(x)=x^{2}$, $x\in R$ is not uniformly continuous on R.

2. Answer any one question: 1 x 5
(a) Let $I=[a,b]$ be a closed and bounded interval and $f:[a,b]\rightarrow R$ be continuous on I. Then prove that $f(I)=\{f(x):x\in I\}$ is a closed bounded interval.
(b) Let $f:[a,b]\rightarrow R$ and $g:[a,b]\rightarrow R$ be continuous on [a, b] and let $[f(a)-g(a)] \cdot [f(b)-g(b)] < 0$. Show that there exists a point c in (a, b) such that $f(c)=g(c)$. Deduce that $cos x=x^{2}$ for some $x\in(0,\frac{\pi}{2})$. 3+2

3. Answer any one question: 1 x 10
(a) i) Let the functions $f:R\rightarrow R$ and $g:R\rightarrow R$ be both continuous on R. Then prove that the set $S=\{x\in R:f(x)=g(x)\}$ is a closed set in R. 4
ii) Explain for continuity the function f defined by $f(x) = lim_{n\rightarrow\infty} \frac{e^{x} - x^{n} sin nx}{1 + x^{n}}$ ($0 \le x \le 1$) at $x=1$. Explain why the function f does not vanish anywhere in $[0,\frac{\pi}{2}]$ although $f(0)f(\frac{\pi}{2}) < 0$. 6
(b) i) Let [a, b] be a closed and bounded interval and $f:[a,b]\rightarrow R$ be continuous on [a, b]. If $f(a).f(b) < 0$ then prove that $f(x)=0$ has at least one root in (a, b). Hence show that any algebraic equation of an odd power with real co-efficients has at least one real root. 5+2
ii) Show that if a function $f:[a,b]\rightarrow R$ is uniformly continuous on (a, b) then it is continuous on (a, b). Is the converse true? Justify. 2+1

UNIT-2

4. Answer any two questions: 2 x 2
(a) Let I be an interval and $c\in I.$ Let the function $f:I\rightarrow R$ be differentiable at c. Then prove that if $k\in R$, kf is differentiable at c and $(kf)'(c)=kf'(c)$.
(b) Prove that $0 < \frac{1}{x}log(\frac{e^{x}-1}{x}) < 1$, $x > 0$.
(c) Show that there is no real number k for which the equation $x^{3}-3x+k=0$ has two distinct roots in (0, 1).

5. Answer any two questions: 2 x 5
(a) Let $I=[a,b]$ and $f:I\rightarrow R$ be differentiable on I. If $f'(a) \cdot f'(b) < 0$ then prove that there exists a point $c\in(a,b)$ s.t. $f'(c)=0$.
(b) State Cauchy mean value theorem and deduce Lagrange mean value theorem from it. Give geometrical interpretation of Lagrange mean value theorem. 1+2+2
(c) Let $f(x)=e^{-\frac{1}{x^{2}}}sin(\frac{1}{x})$ when $x \ne 0$ and $f(0)=0$. Show that at every point f has a differential coefficient and this is continuous at $x=0$. 5

UNIT-3

6. Answer any two questions: 2 x 2
(a) Use Taylor's theorem to prove that $cos x \ge 1 - \frac{x^{2}}{2}$ for $-\pi < x < \pi$.
(b) Examine if f has a local maximum or a local minimum at 0 where $f(x)=x-[x]$.
(c) Find $\theta$, if $f(x+h)=f(x)+hf'(x)+\frac{h^{2}}{2!}f''(x+\theta h):0 < \theta < 1$ and $f(x)=x^{3}$.

7. Answer any one question: 1 x 10
(a) i) State and prove Taylor's theorem with Lagrange form of remainder. 2+4
ii) If $f(x)=sin x$ prove that $lim_{h\rightarrow0} \theta = \frac{1}{\sqrt{3}}$ where $\theta$ is given by $f(h)=f(0)+hf'(\theta h)$, $0 < \theta < 1$. 4
(b) i) State and prove Maclaurin's infinite series of a function f. 5
ii) Derive infinite series expansion of the function $log(1+x)$, $x > -1$. 5

UNIT-4

8. Answer any three questions: 3 x 2
(a) Define separable metric space with example.
(b) Let (X, d) be a metric space. Prove that a non empty open subset G can be expressed as a union of open balls.
(c) Let $X=R^{2}$, the set of all points in the co-ordinate plane. For $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ in X define $d(x,y)=max\{|x_{1}-y_{1}|, |x_{2}-y_{2}|\}$. Show that $d(x,y)$ is a metric space.

9. Answer any one question: 1 x 5
(a) Define closure of a set S in a metric space. Prove that in any metric space closure of a set S is a closed set.
(b) Let X be the set of all real valued continuous functions defined on the closed interval [a, b]. If for $x, y \in X$, we define $d(x,y)=sup_{a \le t \le b}|x(t)-y(t)|$. Then prove that (X, d) is a metric space. 5