Real Analysis Show that if is uniformly continuous on , then is continuous on , too. Then, explain about the converse. *prove using real analysis We were unable to transcribe this imageSCR We were unable to transcribe this imageWe were unable to transcribe this image
Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if...
Definition: A function f : A → R is said to be uniformly continuous on A if for every e > O there is a δ > 0 such that *for all* z, y € A we have Iz-vl < δ nnplies If(r)-f(y)| < e. In other words a function is uniformly continuous if it is continuous at every point of its domain (e.g. every y A), with the delta response to any epsilon challenge not depending on which point...
Show that is not uniformly continuous on . f:R +R, f(1) = x + 2.0 We were unable to transcribe this image
Iff(x): x and gx)sin x, show that both f and g are uniformly continuous on R, but that their product fg is not uniformly continuous on R.
Real analysis 2. Consider the following three definitions: A function f : R-+R is lax-continuous at a E R provided for all e > 0 there is a 6 > 0 such that for all r E R, if x - al6 then |f(x)- f (a)e A function f : R - R is e-continuous at a E R provided for all e >0 there is a 6 > 0 such that for all r E R, if |a- a...
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous on D. (You do not need to write this down. This only serves as a hint for next parts.) (b) Use (a) and the mean value theorem to prove f(x) = e-% + sin x is uniformly continuous on (0, +00). (c) Use the negation of (a) to prove f(x) = x2 is not uniformly continuous on (0,0).
Mathematical Analysis help 2. If J is uniformly continuous on a bounded subset A of R, prove that f is bounded on A.
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q. II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
Prove that f(x) = is uniformly continuous on (1,00) and not uniformly continuous on (0,1). (19 pts)