WebFeb 21, 2024 · Homework Statement Let f be a continuous function lR (all real numbers) --> lR such that f (x+y) = f (x) + f (y) for x, y in lR. prove that f (n) = n*f (1) for all n in lN (all natural numbers) Homework Equations f is continuous also note and prove that f (0) = 0 The Attempt at a Solution Edit: WebJan 4, 2015 · A function f : X → Y is continuous if, for every x ∈ X and every open set U containing f (x), there exists a neighborhood V of x such that f (V) ⊂ U. Proof: Let C be a closed subset of Y, s.t, C ⊂ Y. Clearly, if C is closed, the set Y-C is open since the compliment of a closed set is an open set (Theorem 6.5).
calculus - Proving that $f(x+y) = f(x) + f(y)$ and $f$ being continuous ...
Web*Response times may vary by subject and question complexity. Median response time is 34 minutes for paid subscribers and may be longer for promotional offers and new subjects. Web2 days ago · Final answer. Transcribed image text: Let the continuous random variables X and 1. At least 5.5 Y be defined by the joint density function f (x,y) = ⎩⎨⎧ 501 0 otherwise for 0 ≤ x,0 ≤ y and x+ y ≤ 10 2. At least 1.5 , but less than 2.5 3. At least 3.5 , but less than 5.5 4. At least 2.5 , but less than 3.5 Determine E[X ∣ Y = 6] 5 ... good morning salutation comma
metric spaces - Prove that if a function $f: X\to Y$ continuous …
WebMay 23, 2015 · The solution I have is that f is not continuous in . (The solution doesn't say more than that.) However, the result I got is that is continuous in . Here's my approach: Lets transform and into their polar coordinates, so that we can approach from any direction by varying : Then is continuous iff By using the polar coordinates and letting we get: WebSo is continuous. (inverse image of closed is closed). This direction only uses compactness of . For the other direction we only need the Hausdorffness of : The diagonal is closed iff … WebAug 16, 2024 · Also, are we to assume that $f (x)$ is continuous? If not, then I don't believe that $f (xy)=f (x)f (y)\implies f (x)=x^c$. Take, for example, any additive non-linear function, $g (x) $ with $g (x+y)=g (x)+g (y)$. Then $f (x)=e^ {g (\log x)}$ satisfies $f (xy)=f (x)f (y)$. Show 6 more comments You must log in to answer this question. good morning rush