Suppose that f : (0, ) R satisfies f(x) - f(y) = f(x/y) for all x,
Question:
a) Prove that f is continuous on (0, ∞) if and only if f is continuous at l.
b) Prove that f is differentiable on (0, ∞) if and only if f is differentiable at 1.
c) Prove that if f is differentiable at 1, then f'(x) = f'(l)/x for all x ∊ (0, ∞).
[If f'(l) = l, then f(x) = log x.]
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 90% (10 reviews)
a Let y n x 0 01 If f is continuous at x 1 then fx 0 fy n ...View the full answer
Answered By
Parvesh Kumar
I am an experienced Mathematics and Statistics tutor with 10 years of experience teaching students and working professionals. I love teaching students who are passionate to learn subjects or wants to understand any mathematics and statistics concept at graduation or master’s level. I have worked with thousands of students in my teaching career. I have helped students deal with difficult topics and subjects like Calculus, Algebra, Discrete Mathematics, Complex analysis, Graph theory, Hypothesis testing, Probability, Statistical Inference and more. After learning from me, students have found Mathematics and Statistics not dull but a fun subject. I can handle almost all curriculum of mathematics. I did B.Sc (mathematics), M.Sc (mathematics), M.Tech (IT) and am also Gate (CS) qualified. I have worked in various college and school and also provided online tutoring to American and Canadian students. I look forward to discussing with you and make learning a meaningful and purposeful
5.00+
4+ Reviews
10+ Question Solved
Related Book For
Question Posted:
Students also viewed these Numerical Analysis questions
-
Suppose that f : [a, b] R is continuous and that for all integers n > 0. a) Evaluate ba P(x) f(x) dx for any polynomial P on R. b) Prove that ba|f(x)|2 dx = 0. c) Show that f(x) = 0 for all x [a,...
-
Suppose that f : R R. If f" exists and is bounded on R, and there is an 0 > 0 such that |f'(x)| > 0 for all x R, prove that there exists a > 0 such that if |f(x0)| < > 0 for some x0 R, then f...
-
a. Suppose that f: (0, 1) R is a non-negative continuous function. Show that (0, 1) exists if and only if lim Є c 1-c f exists. b. Let An = [1 - 1/2n, 1 - 1/2n +1] Suppose that f: (0, 1) R...
-
Mary, who is single, had the following items for 2016: Required: Determine Marys adjusted gross income for 2016. Salary $80,000 Loss on sale of 8 1244 small business stock acquired 3 years ago...
-
In Problem 18, what long-term interest rate would represent a break-even point between using short-term financing as described in part a and long-term financing? (Hint: Divide the interest payments...
-
A biological model measures the production of a certain type of white blood cell (granulocytes) by the function where A and B are positive constants, the exponent m is positive, and x is the number...
-
Hot water at \(50^{\circ} \mathrm{C}\) is routed from one building in which it is generated to an adjoining building in which it is used for space heating. Transfer between the buildings occurs in a...
-
Orange, Inc., sells a LearnIt-Plus software package that consists of their normal LearnIt math tutorial program along with a one-year subscription to the on-line LearnIt Office Hours virtual...
-
At January 1, 2024, Caf Med leased restaurant equipment from Crescent Corporation under a nine-year lease agreement. The lease agreement specifies annual payments of $26,000 beginning January 1,...
-
Suppose that it is one year after EBVs investment in Newco, and Tall-tree makes a Series B investment for 6M shares of Newco at $0.2 per share. Following the Series B investment, what percentage of...
-
Define f on R by Find all n N such that f(n) exists on all of R. r 0 f(x) := 120
-
Let I be an open interval, f : I R, and c l. The function f is said to have a local maximum at c if and only if there is a > 0 such that f(c) > f(x) holds for all |x - c| a) If f has a local...
-
Use the methods introduced in Section 7.1 to evaluate the following integrals. dx Vx + 4
-
At which stage of the new product development process is a physical version of the market offering manufactured? A . Idea generation B . Test marketing C . Marketing strategy development D . Concept...
-
Problem 5 What is the charge (magnitude and sign) of a 2 gram particle if it is stationary in a downward electric field of magnitude 650 [N/C]? E = (650 [N/C])(-) FE F = (mg) (-y) 8
-
Roman military commanders a. such as Marius refused to change the recruitment system. O b. became deeply involved in politics after 133 B.C.E. O O d. always deferred to their superiors in the Senate....
-
5- Calculate the diffraction angles (2 0 ) of beta brass polycrystal alloy from 2 0 = 0 to 2 0 = 130. Crystal structure of beta brass given below is cubic. Chemical composition of beta brass alloy is...
-
A currently accepted or popular style in a given field is a OA. aberration OB. type OC. trend OD. fashion O E. fad ...
-
In nine states, the consumption of marijuana is permitted for recreational use in addition to medical use. If a company in one of these nine states has a zero-tolerance drug policy, should it make...
-
Catalytic hydrogenation of naphthalene over PdC results in rapid addition of 2 moles of H 2 . Propose a structure for this product.
-
The probability that your call to a service line is answered in less than 30 seconds is 0.75. Assume that your calls are independent. (a) If you call 10 times, what is the probability that exactly 9...
-
Continuation of Exercise 3-108. (a) What is the probability that you must call four times to obtain the first answer in less than 30 seconds? (b) What is the mean number of calls until you are...
-
Continuation of Exercise 3-109. (a) What is the probability that you must call six times in order for two of your calls to be answered in less than 30 seconds? (b) What is the mean number of calls to...
-
You observe the price of a latte changes from $4 to $5. Instructions: Round your answer to one decimal place. If you are entering a negative number include a minus sign. a. What is the percentage...
-
Layer 2 switches work at a simple level for communication and switching. With technology advancing and routers being able to make more complex decisions, why is this method still used?
-
Explore the significance of intermedial allusions in multimedia storytelling, such as in graphic novels or hypertext fiction, and their implications for narrative coherence and interpretation.
Study smarter with the SolutionInn App