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MMT 4: Real Analysis

Title Name	IGNOU MMT 4 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 4
Subject Name	Real Analysis
Year	2026
Session	-
Language	English Medium
Assignment Code	MMT 4/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT004 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Format	Ready-to-Print PDF (.soft copy)

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MMT 4 2025 - English

Course Code: MMT-004

Assignment Code: MMT-004/TMA/2025

Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answers.

a) The outer measure m* of the set $A={xinmathbb{R}:x^2=1}cup{-3,2} ext{is}0.$

b) A finite subset of a metric space is totally bounded.

c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.

d) The surface given by the equation x+y+z-sin(xyz) = 0 can also be described by an equation of the form z = f(x, y) in a neighbourhood of the point (0,0).

e) A real valued function f on [a,b] is continuous if it is integrable on [a,b].

2. a) Find the interior, closure, the set of limit points and the boundary of the set

$A={(x,y)inmathbb{R}^2:y=1}$

in R² with the standard metric.

b) Consider $f:mathbb{R}^3 omathbb{R}^3 ext{given by}$

f(x,y,z) = (2x+3y+z,xy,yz,xz)

Find ƒ (2,0,-1).

c) Does Cantor's intersection theorem hold for the metric space X = (0,1] with the standard metric? Justify your answer.

3. a) Obtain the second Taylor's series expansion for the function given by

$f=chi_{[0,1]}+2chi_{[3,5]}+chi_{[6,9]}$

b) Find the Lebesgue integral of the function f given by

$f=chi_{[0,1]}+2chi_{[3,5]}+chi_{[6,9]}$

c) Find and classify the extreme values of (x, y) = xy Subject to the constraint

$frac{x^2}{8}+frac{y^2}{2}-1=0.$

4. a) $f:mathbb{R}^3setminus{(0,0,0)} omathbb{R}^3setminus{(0,0,0)}$ be given by

$f(x,y,z)=left(frac{x}{x^2+y^2+3^2},frac{y}{x^2+y^2+z^2},frac{z}{x^2+y^2+z^2} ight)$

Show that f is locally invertible at all points in $mathbb{R}^{3}$ {(0,0,0)}.

b) For the equation , x² + y³ + z³ = at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form z = f (x, y) .

c) Find the Fourier series of f (t) = t² on [−π,π].

5. a) Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set D U E where

$D={(x,y)inmathbb{R}^2:||(x+1,y)||=1} ext{and}$

$E={(x,y)inmathbb{R}^3:||x-1,y||=1}.$

b) Which of the following subsets of R are compact w.r.t. the metric given against them. Justify your answer.

i) A = (1,0) in $mathbb{R}$ of − $mathbb{R}$ with standard metric.

ii) A = [4,3] − $mathbb{R}$ with discrete metric.

iii) {(x, y) ∈ $mathbb{R}^{2}$ y > 0} − $mathbb{R}^{2}$ with standard metric

6. a) If E is a subset of $mathbb{R}$ with standard metric, then show that $(overline{E})^C=(E^C)^0.$

b) Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

c) Find the interior and closure of the set $mathbb{Q}$ of rationals in $mathbb{R}$ with standard metric.

7. a) Let F be the function from $mathbb{R}^{2}$ to $mathbb{R}^{2}$ defined by

F(x, y) = (x² + y² , xy)

Show that F is differentiable at (2,1) . Find the differential matrix of F.

b) Show that the function f defined by

$f(x,y)=egin{cases}frac{xy}{x^2+y^2},& ext{if}(x,y) e(0,0)\0,& ext{if}(x,y)=(0,0)end{cases}$

is not differentiable at (0,0). Does the partial derivatives of f exists at (0,0)? or any at any other point in R²? Justify your answer.

c) Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

8. a) Find the directional derivative of the function $f:mathbb{R}^{4} ightarrowmathbb{R}^{4}$ defined by

$f(x,y,z,w)=(x^2 y,xyz,x^2+y^2,zw^2)$

at the point (1,2,-1,-2) in the direction v = (1,0,-2,2).

b) Suppose that $f:mathbb{R} omathbb{R}^2$ is given by f(t) = (t,t²) and $g:mathbb{R}^2 omathbb{R}^3$ is given by g(x, y) = (x², xy, y²-x²). Compute the derivative of gof.

c) Find $Bleft[2,frac{1}{2} ight]$ in $mathbb{R}$ where d is the metric given by $d(x,y)=min{1,|x-y|}.$

9. a) Give an example of a family f₁ of subsets of a set X which has finite intersection property. Justify your choice of example.

b) Verify the hypothesis and conclusions of the Fatou's lemma for the sequence {f_n} given by

$f_n(x)=2n ext{for}xinleft(frac{1}{2n},frac{1}{n} ight)$

$=0quad ext{for}xinleft(0,frac{1}{2n} ight)cupleft(frac{1}{n},1 ight)$

10. a) Let (X,d) be a metric space and A be a subset of X. Show that bdy(A) = Qif and only if A is both open and closed.

b) Give an example of an algebra which is not a σ − algebra. Justify your choice of examples.

c) If E is a measurable set and f is a simple function such that a $ale f(x)le bquadforall xin E,$ , show that

$am(E)leint_E f,dmle m(E)b.$

MMT 4 2026 - English

Assignment (MMT – 004)

Course Code: MMT-004
Assignment Code: MMT-004/TMA/2026
Maximum Marks: 100

1. State whether the following statements are true or false. Justify your answers. $5 \times 2 = 10$

a) The outer measure m^ of the set $A = \{x \in \mathbb{R} : x^2 = 1\} \cup [-3, 2]$ is 0.

b) A finite subset of a metric space is totally bounded.

c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.

d) The surface given by the equation $x + y + z - \sin(xyz) = 0$ can also be described by an equation of the form $z = f(x, y)$ in a neighbourhood of the point (0, 0).

e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].

2. a) Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

b) Consider $f : \mathbb{R}^3 \to \mathbb{R}^3$ given by

$$f(x, y, z) = (2x + 3y + z, xy, yz, xz)$

Find f(2, 0, -1).

c) Does Cantor’s intersection theorem hold for the metric space $X = (0, 1]$ with the standard metric? Justify your answer.

3. a) Obtain the second Taylor’s series expansion for the function given by

$$f(x, y) = xy^2 + 5xe^y \text{ at } \left(1, \frac{\pi}{2}\right).$

b) Find the Lebesgue integral of the function f given by

$$f = \chi_{[0, 1]} + 2\chi_{(3, 5]} + \chi_{[6, 9]}.$

c) Find and classify the extreme values of $(x, y) = xy$

Subject to the constraint

$$\frac{x^2}{8} + \frac{y^2}{2} - 1 = 0.$

Based on the image provided, here is the transcription of the text:

4. a) Let $f : \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}^3 \setminus \{(0,0,0)\}$ be given by

$$f(x, y, z) = \left( \frac{x}{x^2 + y^2 + z^2}, \frac{y}{x^2 + y^2 + z^2}, \frac{z}{x^2 + y^2 + z^2} \right)$

Show that f is locally invertible at all points in $\mathbb{R}^3 \setminus \{(0,0,0)\}$ .

b) For the equation $x^2 + y^2 + z^3 = 0$ , at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form $z = f(x, y)$ .

c) Find the Fourier series of $f(t) = t^2$ on $[-\pi, \pi]$ .

5. a) Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set $D \cup E$ where

$$D = \{(x, y) \in \mathbb{R}^2 : \|(x+1, y)\| = 1\} \text{ and}$

$$E = \{(x, y) \in \mathbb{R}^3 : \|x-1, y\| = 1\}.$

b) Which of the following subsets of $\mathbb{R}$ are compact w.r.t. the metric given against them. Justify your answer.

i) $A = (0, 1)$ in $\mathbb{R}$ of $-\mathbb{R}$ with standard metric.
ii) $A = [3, 4] - \mathbb{R}$ with discrete metric.
iii) $\{(x, y) \in \mathbb{R}^2 \quad y > 0\} - \mathbb{R}^2$ with standard metric.

6. a) If E is a subset of $\mathbb{R}$ with standard metric, then show that $(\bar{E})^c = (E^c)^\circ$ .

b) Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

c) Find the interior and closure of the set $\mathbb{Q}$ of rationals in $\mathbb{R}$ with standard metric.

7. a) Let F be the function from $\mathbb{R}^2$ to $\mathbb{R}^2$ defined by

$$F(x, y) = (x^2 + y^2, xy).$

Show that F is differentiable at (1, 2). Find the differential matrix of F.

b) Show that the function f defined by
$$f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & \text{if } (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases}$
is not differentiable at (0,0). Do the partial derivatives of f exist at (0,0)? or at any other point in $\mathbb{R}^2$ ? Justify your answer.

c) Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

8. a) Find the directional derivative of the function $f : \mathbb{R}^4 \to \mathbb{R}^4$ defined by

$$f(x, y, z, w) = (x^2y, xyz, x^2 + y^2, zw^2)$

at the point (1, 2, -1, -2) in the direction $v = (1, 0, -2, 2)$ .

b) Suppose that $f : \mathbb{R} \to \mathbb{R}^2$ is given by $f(t) = (t, t^2)$ and $g : \mathbb{R}^2 \to \mathbb{R}^3$ is given by

$g(x, y) = (x^2, xy, y^2 - x^2)$ . Compute the derivative of $g \circ f$ .

c) Find $B \left[ 2, \frac{1}{2} \right]$ in $\mathbb{R}$ where d is the metric given by $d(x, y) = \min\{1, |x - y|\}$ .

Based on the image provided, here is the transcription of the final questions:

9. a) Give an example of a family f_i of subsets of a set X which has finite intersection property. Justify your choice of example.

b) Verify the hypothesis and conclusions of the Fatou’s lemma for the sequence $\{f_n\}$ given by

$$f_n(x) = 2n \text{ for } x \in \left( \frac{1}{2n}, \frac{1}{n} \right)$

$$= 0 \text{ for } x \in \left( 0, \frac{1}{2n} \right] \cup \left[ \frac{1}{n}, 1 \right]$

10. a) Let (X, d) be a metric space and A be a subset of X. Show that $bdy(A) = \emptyset$ if and only if A is both open and closed.

b) Give an example of an algebra which is not a $\sigma$ -algebra. Justify your choice of examples.

c) If E is a measurable set and f is a simple function such that $a \leq f(x) \leq b \text{ } \forall x \in E$ , show that

$$am(E) \leq \int_{E} f \, dm \leq m(E)b.$

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