Q) check the continuity of X) at X-1, 0,1.23 (x²-1, 2x, -1

Q: Examine the continuity of f(z) = 2-√z+4 Z at z = 0

A: Given: fz=2-z+4z We have to find the continuity of f(z) at z=0.

Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

A: A function is said to be bijective if the function is both one one and onto. The given function is…

Q: Prove that f(z)= 1/(1-z) in not uniformly continuous for |z|

A:

Q: 4. Use the definition of uniformly continuity to prove that f (x) 1- x is uniformly continuous on…

A:

Q: 21. Discuss the continuity of : 1 for x+- x 2 (ans.: discontinuous at x=0,2 ; continuous at x=1)

A: Given: f(x)=x+1x   for x<0-x3      for 0≤x<1-1       for 1≤x<21          for  x=20…

Q: Q5: State and prove the theorem of (continuity-differentiability), then find the value(s) of x that…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: ɔ that h is a continuous functi (x2 — 2, х < 6 h(x) 3= 7х + a, х 6

A:

Q: Using the e, d definition of uniform continuity: Prove that f(x) = x² is not uniformly continuous on…

A:

Q: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

A: Prove that all points in the open interval (0,1) are limit-points (we are working on the real line)

Q: Show that f(x, y) = point (5,-3). 3x+2y x+y+1 meets the three conditions for continuity at Add-ins…

A:

Q: 6. Use definition of continuity to determine the value(s) of k such that the 2х + 5, х< 3 function f…

A: Topic : Continuity

Q: lim r" use Cauchy Criteria to prove that exists and 0<r<1 n → 00

A: To Prove - Use Cauchy Criteria to prove that lim rn exists and 0 ≤ r &lt; 1 , n →∞         Theorem…

Q: Using continuity prove that f : (0, ∞) → R  defined by ƒ(x) : = 1/x is continous.

A:

Q: Suppose that ƒ″ is continuous on [a, b] and that ƒ has three zerosin the interval. Show that ƒ″ has…

A: It is given that, f'' is continuous on a,b and f has three zeroes in a,b. To show: f'' has at-least…

Q: check the continuity of f (x) at x 0, 1, 2, 3 and 4 2- x, 0sx <1 2, 3, 2x - 2, 2<x 53 (10-2x, 3<x S4…

A:

Q: Q1/ Discuss the continuity of : 1 x+ for x 2

A: Since, you have post multiple question. I am answering first one. Please post again for remaining…

Q: 5 - x, -1 sx 2 r - 1, 2 <xs 3 Discuss the continuity of g(x) = %3D

A:

Q: Show that x is continuous everywhere. r(x)-x-0 XX<0

A:

Q: Suppose f is continuously differentiable on an interval (a, b). Prove that on any closed subinterval…

A: We will use the continuity &amp; differentiability property of f inside the interval [x , x0] ,…

Q: Prove that the function ƒ : (0, ∞) → R defined by ƒ(x) = x³/² = (√x)³ is not uniformly continuous.…

A: To Prove: The function f:0, ∞→ℝ defined by fx=x3/2=x3 is not uniformly continuous. To Use the…

Q: (d) Consider the set [1, √5] C [-1, √10]. Explain f is not continuous by (i) connectedness, (ii)…

A:

Q: 23. Suppose that the functions f and g are defined throughout an open interval containing the point…

A:

Q: Use Euler’s Theorem to find the degree of homogeneity for f( x1 , x2) = x3 + y3 + 3xy2

A: We will use the Euler's theorem which is : x•fx + y•fy = nf  Where, n = degree of homogeneity    The…

Q: (5) Prove that Lp(f)<Uq(f) for any bounded function f and partitions P,Q of [a, b]. (Hint: Consider…

A: Note: According to Bartleby guidelines; for more than one question asked only the first one is to be…

Q: (b) Let U, T be usual and cofinite topologies on se:t of real numbers R and f be a map from R into R…

A: As per our guidelines we are supposed to answer only first question. So i have solve  question (b)…

Q: b. Test the continuity of the following function 1, x< 0 V1-x, 0sxs1 ,atx 0,1 x- 2, x21

A: We have to check the continuity of f(x) at x=0, 1

Q: Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if f: (a,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Q2) Discuss the continuity at x=2. When: x> 2 x = 2 3x + 1 f(x) : x² + 3 %3D 2x + 1 x< 2

A: The given functionf(x)=3x+1x&gt;2x2+3x=22x+1x&lt;2to check continuity at x=2

Q: Prove Fermat’s Theorem for the case in which f has a localminimum at c .

A: It is given that f has a local minimum value at c. Thus, by the concept of local minimum, fc≤fx when…

Q: Show that x| is continuous everywhere. r(x)- x- 0 xX <0

A: f(x) is continuous, if LHL=RHL=f(x&gt;0), where LHL is left hand limit while RHL is right hand…

Q: Show using Definition 4.3.1(Continuity) that if c is an isolated point ofA ⊆ R, then f : A → R is…

A: If c∈A⊆ R is an isolated point of A ⊆ R. To prove  f : A → R is continuous at c. Let ∈&gt;0 , Since…

Q: 5. Determine the interval on which f(x) = x+2 is integrable. %3D (-00, 2) [-2, co0) (-00,-2) U (-2,…

A:

Q: x + 3, x 2 x - 2

A:

Q: Using the definition of continuity directly prove that f : R → R defined by f(x) := x2 is continuous

A: Using the definition of continuity, the given question has been proved in step 2.

Q: 3) Consider the topology 7 = {G CR: 0 ¢ G or G = R} on R. Is open T 3= interval (-1,1) compact in…

A:

Q: 2. (a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

Q: Define f(x) = x² for all x. Verify the e-8 criterion for continuity at x = 2.

A: Given real valued function  f(x) = x2, ∀x We have to show that the given function is continuous at x…

Q: Q2) Discuss the continuity at x=2. When: x> 2 X = 2 x< 2 Зх + 1 f(x) = } x? + 3 %3D 2х + 1

A: Our objective is to discuss the continuity at x=2.

Q: Of f is a posittue differentiable furnction on. (00). (? (a+nx) ) Arone existo finitely .and…

A:

Q: Consider the topology 7 = {GCR:0 ¢G or G = R} on R. Is open interval (-1, 1) compact in (R, T)?…

A:

Q: 2. Use the definition of uniform continuity to show that any Lipschitz function f on E is uni-…

A:

Q: - Using prove M the E - f ? f(x) continuous 8 definition of continuity R > R that is defined by 3x +…

A:

Q: Use the Mean Value Theorem to prove that, for all x, y € IR, cos x - cos y ≤|x-y.

A:

Q: Suppose that f(x) is convex on an interval I. Show that f(x1) + f(x2) + ·+ f(xn) n for arbitrary n…

A:

Q: Let X and Y be normed linear spaces and F : X → Y be linear. Prove that F is continuous if and only…

A: Let X and Y be matric space and let f: x→y be a uniformly continuous function. If ( xn)n∈N is a…

Q: Show that the space C(R) of all continuous functions defined on the real line is an…

A: According to the given information, it is required to show that C(R) of all continuous functions…

Q: 2. Define f : (2,7) → R by f (x) = x³ – x + 1. Use the definition of uniformly continuity to prove…

A: Consider the given information. fx=x3-x+1 As the function is in the form of a polynomial.  Now, find…

Q: 3. Use Cauchy's residue theorem to evaluate the function f(2) = z around (z– the circle |z = 2 in…

A:

Q: Let f,g: D -> R be conitnuous at c ∈ D. Prove that fg is continuous at c.

A: