Define f : R" - ||x ||<) +7² _if ||x || < 1 and f(x) [0, 0) by letting f(x) = if ||x|| > 1. Prove that f is continuous on R". (To prove that f is continuous at a given xo E R™, consider separately the following three cases: ||x0|| > 1, ||x0|| < 1, ||o|| = 1.) = 0

Q: Prove: f(x)=5x is continuous at x-2

A: Given function is fx=5x,

Q: If f has a removable discontinuity at x = c, then it is possible to redefine f ( c) so that f is…

A: When a function has a removable discontinuity at a point, it means that the function is not defined…

Q: Suppose f is a function defined and continuous on the open interval (a, b). Suppose g is a function…

A: Given that: f is a function defined and continuous on the open interval (a, b).g is a function…

Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: True False Suppose f(x) is continuous everywhere. Must f(x) attain an absolute maximum somewhere on…

A: Extreme value theorem states that if f(x) is continuous on closed interval I, then f(x) attains…

Q: Define f : R" → [0, 0) by letting f(x) = (1 – ||æ|| 1. Prove that f is continuous on R™. (To prove…

A: As per the instruction, we will be using the Lemma 3 (which is written in the question) to show that…

Q: Let 1-tan r f(x)= ExtIE 0,5], 1f f(x) is continuous in [0.), then is 1 -1 /2

A:

Q: Show that if f is analytic in a region G and if the range of f lies in a straight line, then f is…

A:

Q: Let 1 f(x) = ,* – 8, – 4 < x < 8 1 The domain of f¯'is the interval [A, B] where A and B = Check…

A:

Q: True/False; f (x) =; 1 is continuous on 2х —1 A) False B) True If f(-3)=6, f(2)= and f is continuous…

A:

Q: 5) Suppose that f is continuous on (-00, 00). How many non-differentiable points could f have? Is it…

A:

Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x&lt;1           x…

Q: This is real analysis question. Recall that if a function f : R → R is differentiable, then it is…

A: Let f: ℝ→ℝ be a differentiable function. (a) Suppose f'x≥1 for all x∈ℝ. Let K⊂ range (f) be a…

Q: Show that f is a continuous function for all x, on the interval -L < x ≤ L f(x) = (x / L) (1 - |x /…

A:

Q: Which of the following choices correctly discusses the continuity of the following function? F(x)=…

A: Given :  F(x) = x+3x2-9   To determine : The continuity of  a function.

Q: What is an example of a function f : [2, 3] -->R which is not continuous at 1 but which is…

A: Here's an example of a function that is not continuous at 1 but is integrable on the interval [2,…

Q: if x ≤ 0 if 0 2 Use the definition for g to evaluate the following. Show work where appropriate. 1,…

A:

Q: s at every number in its domain. 2x+1, f(x) = if x 5

A: Given : f(x)=2x+1      if x≤3ax+b     if 3&lt;x&lt;5x2+2     if x≥5      To Find : Value of a and b…

Q: Consider the function f with domain (-2, : 3|22 – 1| if – 2 < x < 1 2x2 + 2x – 4 f(x) = 3 if x = 1…

A:

Q: 2. Let f be continuous and differentiable everywhere. Suppose that f(-1) = f(1). Show that there are…

A:

Q: Let f. ROR be a Continuous function with f(x) ≤ 10 for = all XER. Explain why f is not homeomorphism

A: A homeomorphism is a continuous bijection with continuous inverse.

Q: Use the definition of continuity to determine if each function is continuous or discontinuous at…

A: Given function:- fx=x-2x≤-1-xx&gt;-1 Condition to continuity at x=alimx→a-fx=limx→a+fx=fa