WebOct 29, 2024 · Question: Let (X, d1) and (Y, d2) be two metric spaces and f, g: X ↦ Y be two continuous functions. Then prove that {x ∈ X: f(x) = g(x)} is closed in X. Approach: We consider the function h: X ↦ R + ∪ {0} defined by h(x) = d2(f(x), g(x)) Lemma 1: h(x) is continuous on X. 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.

Provided $f$ is continuous at $x_0$, and $f(x+y) = f(x)

WebA: given f(x)=x centered at x=4 f(x)=x f(4)=2f'(x)=12x-12… question_answer Q: Find the directional derivatives of the following functions at the specified point for the specified… WebViewed 3k times 2 Consider the function f: R 2 → R given by f ( x, y) = max ( x, y). (That is, f ( x, y) is the larger of x and y, so f ( − 3, 2) = 2, f ( 1, 4) = 4, and f ( − 3, − 2) = − 2 .) (assume that R 2 has sup metric) prove that f is continuous. how to take the jlpt n5

Proof of continuous. f(x+y)=f(x)+f(y) Physics Forums

WebSo now we see that if ( x n, y n) ∈ G ( f), ( x n, y n) → ( x, y), then y n → f ( x n) as defined by G ( f) and x n → x, f ( x n) → y. Since f is assumed to be continuous, f ( x n) → f ( x) so y = f ( x). Therefore ( x, y) ∈ G ( f) and we conclude G ( f) is closed. Share Cite Follow edited Feb 22, 2016 at 22:06 YoTengoUnLCD 13.1k 4 39 99 WebOct 5, 2024 · Let f ( x, y) be a continuous real-valued function on the unit square [ 0, 1] × [ 0, 1]. Show that h ( x) = max { f ( x, y): y ∈ [ 0, 1] }, is also continuous. Answer. Since f ( x, y) is continuous, then max { f ( x, y) } is also continuous on [ 0, 1] × [ 0, 1]. Thus for any fixed values of y ∈ [ 0, 1] , max { f ( x, y) } is also continuous . reagan reborn wasteland 3