Show that (1) => (2) and (2) => (1). (1) X is connected. (2) There is no continuous function f : X → {-1,1}, where {-1,1} is equipped with discrete topology. Note: The function need not be surjective.

Q: Suppose f: (0, 1)→ (0, 1) is a surjective monotonic function. True or false: The function f is…

A: f: 0,1→0,1 is a surjective monotonic function.

Q: Let f:R-->R and suppose that |f(x)-f(y)| less than or equal to |x-y|^a for some a>1. Show that f…

A: Let f:R-->R and suppose that |f(x)-f(y)| less than or equal to |x-y|^a for some a>1. Show that…

Q: Is there a real function f(x) that is differentiable only at a point x?

A:

Q: 4) (1, 4) 2) (3.-4) (-4, 5) Determine if f is continuous at x = c. (a) x = 4 (b) x = 1 (c) x = -1…

A: Given   The graph of the function  Definition of Continuity :-  A function f is continues at x= c…

Q: Define g(-5) in a way that extends g(x)= 9(-5) = x² - 25 x + 8x + 15 to be continuous at x = -5.

A: We have to define g(-5) such that g(x) is continuous at x = -5.

Q: Determine which of the points in the following list lie in the domain of the function f(x, y) =…

A: Let's find.

Q: Suppose that f(x) is a continuous function with f(-3) = – 4 and f(1) = 4. Determine which choice…

A: Consider a function f(x) defined on an interval a,b. Suppose that f is continuous and f(a)≠f(b).…

Q: Sketch a graph of a single continuous function with a y-intercept at ( 0 , 2 ) that has the…

A: Given, f(x) is a continuous function satisfying the following table.   -∞,-6 -6, 1 1, 3 3, ∞…

Q: f() is decreasing at r = -5 f (z) has a local minimum at z = -2 f(z) has a local maximum at r = 2

A:

Q: Suppose that f(x) is a continuous function with f(3)=−7 and f(7)=7Determine which choice best…

A: The given statement is For some c in the interval [-7,7], f(x)=c for all x in the interval [3, 7]…

Q: Let f be a function defined on [15, 23]. Suppose for every x ≤ (15, 23) d da If f(16) = 44, find the…

A: We  know By definition of derivative  in interval a,b⇒f'x=limx→afx-fax-a

Q: Find the values of a and b that make f continuous everywhere. b = f(x): x-2 ax² - 6x + 3 4x -a + b…

A:

Q: Show that the function f : [0, 1] →[0, 1] given by f(x) = (x −1)2 has a fixed point. Can you apply…

A:

Q: Let f be the function defined by ƒ (x) = x² − 12x³ + 48x² − 64x. Which interval is the graph of f…

A:

Q: Suppose that f(x) is a continuous function with f( – 1) = - 8 and f(3) = 8. Determine which choice…

A: Always true

Q: x3 – 7x2 + 5x + 7 = 0, (-1, 1) Let f(x) = x3 - 7x2 + 5x + 7. f(x) is ---Select--- f(-1) = on the…

A:

Q: Construct a continuous function that shows that the intervals [0,1] and [2,6] are homeomorphic

A: In the given question we have to find a continuous function from the interval [0,1] to the interval…

Q: Let f be a function defined on [11, 25]. Suppose for every x = (11, 25) d. dx -f(x) = 0. If f(12) =…

A: Given query is to find the value o f(24).

Q: Suppose that f(x) is a continuous function on the interval [0, 4] with f(0) = -10 and f(4) = 10.…

A:

Q: Consider the function f(x) = 2x³-32²-362-6 on the interval [-3, 0]. f(x) is continuous on [-3,0], so…

A:

Q: Describe the interval(s) on which the function is continuous. (TX) f(x) = sec

A: f(x)=sec⁡(πx4)

Q: Suppose that f(E) is a continuous function with f(– 4) = – 9 and f(0) best describes the following…

A:

Q: Let f(x) be the function x + 5 for x 2.

A: Let f(x)=x+5 for x<-1               =cx for -1≤x≤2                =x+6 for x>2 To find the…