Department of Mathematics Guru Nanak Dev University, Amritsar-143005 Assignment #2: B.Sc.(Hons) Physics Semester I MTL 131: Continuity, Differentiability July 28, 2011 Definition 1: f (x) is said to be continuous at a point x = a of its domain if for every ϵ > 0, there is a δ > 0 such that |x − a| < δ ⇒ |f (x) − f (a)| < ϵ Definition 2: f (x) is said to be differentiable at a point x = a of its domain with derivative f ′ (a) ∈ R if for every ϵ > 0, there exists a δ > 0 such that f (x) − f (a) ′ |x − a| < δ ⇒ − f (a) < ϵ x−a Remark 1: If f is continuous at x = a, we write lim f (x) = f (a). x→a
f (x) − f (a) = f ′ (a). x−a Theorem 1: If a function f is differentiable at a point x = a with derivative f ′ (a) then f is continuous at x = a. Proof: Let f is differentiable at a point x = a with derivative f ′ (a). Then for every ϵ > 0 there is a δ > 0 such that f (x) − f (a) |x − a| < δ1 ⇒ − f ′ (a) < ϵ x−a Remark 2: If f is differentiable at x = a with derivative f ′ (a), we write lim
x→a
f (x) − f (a) − |f ′ (a)| ≤ f (x) − f (a) − f ′ (a) < ϵ, x−a x−a
As
we need to find a δ > 0 such that |x − a| < δ ⇒ |f (x) − f (a)| ≤ (ϵ + |f ′ (a)|)|x − a| < ϵ. { } ϵ This can be done by defining δ := min δ1 , . Result follows now. ϵ + |f ′ (a)| Remark 2: Converse of the theorem 1 is not true as there are continuous functions which are not differentiable. An example is f (x) = |x| which is continuous at x = 0 but fails to have derivative at x = 0 as can be seen readily from the |x| − 0 |x| following: lim = lim which does not exist! x→0 x − 0 x→0 x 1. Check the continuity of the following functions: (a) f (x) := x − |x| at point x = 2. |x| at x = 0. x ( ) 1 x sin if x ̸= 0 (c) f (x) := at x = 0. x 0 if x = 0
(b) f (x) :=
x − 1 if x > 1 2 + x if x < 1 at (d) f (x) := 1 if x = 1 −a + bx 8 2. If the function f (x) = b − ax
x = 1.
2 x − 4x + 4 if x > 2 x2 − 4 (e) f (x) := , at x = 2. 2 if x < 2 (x − 2) 0 if x = 2 sin 2x if x > 0 x (f) f (x) := , at point x = 0. x e − 1 2 if x ≤ 0 x { sin 4x if x > 0 (g) f (x) := , at point x = 0. sin 3x 0 otherwise
if x < 1 if x = 1 ; is continuous at x = 1 what are the possible values of a and b? if x > 1
3. Let f and g be two real functions. If f and g are continuous at a point x = a, then prove that so are f (x) ± g(x) and f (x)g(x). Hence conclude that the functions x + sin(2x) and x sin(ax) are continuous at all x ∈ R. 4. If f (x) := x + 2 and g(x) := 2 − x2 , evaluate (f ◦ g)′ (1) and (g ◦ f )′ (1). ( ) 1 2 if x ̸= 0 x sin is differentiable at x = 0? 5. Check whether the function f (x) := x 0 if x = 0 6. Check differentiability of the function f (x) := |x − 2|3 at x = 2.
Jul 28, 2011 - ... to be continuous at a point x = a of its domain if for every ϵ > 0, there is a δ > 0 such that ... Check the continuity of the following functions:.
Department of Mathematics, University of Scranton, Scranton, PA 18510,. USA. CNRS, I3S, ESSI, BP 145, Route des Colles,. 06 903 Sophia Antipolis, France. 1 ...
Jan 31, 2010 - after I obtained the masters degree. Manuscripts DâH ...... Then the dual (Ft)tâ¥0-predictable projection of (At)tâ¥0 is for t ⥠0 given by. Ap t = â«.
Part B Measure angle B and angle C to the nearest degree. On the lines .... Which of these graphs best shows the relationship between time and the number of.
theme. Serge Tabachnikov. â. Department of Mathematics, Penn State University. University Park ... in the ambient plane do not change; such curves are called.
Applications are invited for one the post of Technical/Secretarial Assistant to work under. UGC-SAP(DRS-I). Candidates should be a B.Sc in Mathematics and capable of good typing speed on the computer. Age limit of candidate is as per UGC rules and ot
Advertisement. Applications are invited for one the post of Technical/Secretarial Assistant to work under. UGC-SAP(DRS-I). Candidates should be a B.Sc in ...
Jan 27, 2012 - BSc/ Masters Degree in Architecture (4years minimum study in design), Drivers .... Degree or B-Tech qualification in Town Planning, Candidate ...
We must find a way to work with other ... I would call a 'no-kidding, ultra-hard mission'. And if you're ... As you consider the need and opportunity for boundary spanning in .... of using a secure video conference system, âI would go to General ..
Capacitor is a storage element which can store and deliver energy in electrical field. ... Vidyarthiplus.in. Inductor is the element in which energy is stored in the form of ..... There are two alternative methods of connection, depending on which ..
One in four adolescents also reports verbal, physical, emotional, or sexual abuse each year. These teens may be at a greater risk for unhealthy and abusive relationships as adults. Working together, men, women, and teens can change attitudes and perc
May 1, 2017 - ... subjected to discrimination in the provision of any care or service of the. ACH on the grounds of race, color, national origin, or handicap.
Feb 10, 2016 - automatic provincial finalists, provided that they meet the basic eligibility requirements of ... 750-0837; 818-5656 or through email addresses.
When a Notices editor asked me to review Badiou's book [2] I objected on the grounds that I am no philosopher, which only strengthened her determination. Here then is a mathematician's review of a philosopher's mathematics book. Alain Badiou (born 19
Learners: Ready to Face Life's Challenges,â we hope we have equipped you with 21st century skills and lifelong competencies through quality education that is ...
Aug 30, 2016 - please contact Rebecca B. Bond, Chief, Disability Rights Section, Civil Rights Division, U.S.. Department of Justice, by any one of the following ...
members in maximizing the efficiency, cost-competitiveness, safety and quality of marine cargo operations in the Port of New York and New Jersey." 3 ...
compliance with statutory population limits. Applicant may reapply if a license becomes available. Continuous tabling of the application denies the applicant their ...