(8) A space X is connected space IFF every continuous funetion f:X → D, Where D-(0,1} with discrete topology is: (a) Constant function (b) Not constant function (c) Don not has the intermediate property (d) Non of above
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Step by step
Solved in 2 steps
- Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.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.Which is not true? If a function is continuous on a closed interval [a,b], then it has a maximum and a minimum on that interval. If a function has a maximum and a minimum over a closed interval, then it is continuous on that interval If a function has no extreme values on [a,b], then it is continuous on that interval. If a function has either a maximum only or a minimum only over a closed interval, then it is discontinuous on that interval.
- Prove Corollary 5.5.10 - Let f be a continuous real-valued function defined on a metric space (X, d), and let D be a compact subset of X. Then f assumes maximum and minimum values on D.8 Examine holomorphism of given function f(z) = |z|2The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that g(x) = 0 for 7 distinct values of x in (a, b).What is the minimum number, k, of z in (a, b) such that g'(z) = 0?
- Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0<=ƒ(x)<= 1 for every x in [0, 1] . Show that there must exist a number c in [0, 1] such that ƒ(c) = c (c is called a fixed point of ƒ).1. Show (in terms of ε - ? ) that a function f : [2 ; 7]→ R be defined by f(x) = square root of (x2 +1) is uniformly continuous.Prove that there is a function g(x) that is not everywehre continuous, yet it is even. Prove that there is a function h(x) that is not everywhere continuous, yet is odd.
- The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that g(x) = 0 for 4 distinct values of x in (a, b).What is the minimum number, k, of z in (a, b) such that g'(z) = 0? k=?8 Examine holomorphism for given function f(z) = |z|Assume f and g are as described in Theorem 5.3.6,(L’Hospital’s Rule: 0/0 case). but nowadd the assumption that f and g are differentiable at a, and f' and g' are continuous at a with g'(a) not equal to 0. Find a short proof for the 0/0 case of L’Hospital’s Rule under this stronger hypothesis.