3.18 Suppose if x = 0, f(x) = if z e (. where n e N. Prove: f € R[0, 1], even though f has infinitely many discontinuities. (Hint: Consider Theorem 3.2.3.)

Q: 1 Let f (x,y) =In Then f (1, – 1) =... 3x2+4y3 XX

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Q: Suppose that f: [0,1]–R is a function satisfying the conclusions of the intermediate value theorem.…

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Q: 3. Show that the functionf on [-1,1] defined as if f (r) = 1+e if x = 0 is discontinuous at x = = 0.…

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Q: Q.No.1.|A f:X→Y where X,Y E Z*, and 02X<23 and, 02 Y<59, X is hour of clock and Y is minute of…

A: Let f:X→Y,  where X,Y ∈ℤ+ and 0≥X≤23,0≥Y≤59. Here, X denote the hour of clock and Y denote the…

Q: 8. Given the following function f, define f(1) so that f is con- tinuous at x = 1, if possible: Зх —…

A: Given: fx=3x-1if x&lt;1x2+1if x&gt;1

Q: If f (x) = x² – 4, what is the simplest form of +1), %3D - X -1 Select one: a. х + 3 о ь. х — 1 c. x…

A: The given function is: f(x)=x2-4

Q: 4.1.7. Show that if f : R" + R" satisfies |f(x)| < |r|² for all r € R" then f is differentiable at…

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Q: Exercise 2. Prove that dim(null T") = dim(null T) + dim W – dim V and dim(range T") = dim(range T),…

A: We prove dim null T*=dim null T + dim W-dim V and dim range T *=dim range T for every T L(V·W).

Q: 3.7 Suppose f and g are in R[a, b] and f(x) < g(x) for all r € [a, b]. Prove: f(x) dr < | g(x) dr.…

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Q: R.4. Let f:R→R be defined by f(x)= 2x² – 3, and let A=[-5,3]. Find f[A]. f'[4]: S[S"[4] , and…

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Q: 5. Let f: NxN → Z be defined by f(m, n) = your answers. What is the preimage of 0? =m-n. Is f…

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Q: Prove that if a(x) = |r|, the absolute value function, then a E C(R). (Use 1.9.)

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Q: Suppose f: C → C is a holomorphic function. Let g(z) = ƒ(₹). Using the Cauchy Riemann equations, or…

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Q: Let f: (0,3] →→→R such that f(x) = 1². • Show that f(r) < 1. • Show that f has no fixed point on…

A: The given function f:(0,13]→ℝ is defined as: fx=x2. We need to show that: fx&lt;x. f has no fixed…

Q: If f(x)= x - 4 and g(x)= Vx-2; which of the following is equal to х — 2 Option 1 – 4(SF 2) X - х — 2…

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Q: 7) Use Theorem 7.1.1 (Table) to find L{f(1}} for the following: S)-(e*-e*)}° a.

A: The given function is f(t)=(e-3t-e5t)2. Expand the function implies, f(t)=e-6t+e10t-2e2t The Laplace…

Q: 3.2 † Prove part (ii) of Theorem 3.1.2. Theorem 3.1.2 Let f and g be in R[a, 6} and c e R. Then i.…

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Q: 10. Consider the following function { 1 if 2n <t <2n +1, n E N -1 if 2n +1<t< 2n + 2, n EN f (t) =…

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Q: Do part C only without PMI.

A: Step 1: Part C. Without using the principle of mathematical induction (PMI). Given the…

Q: 3.19 Suppose if r = 0, f(x) = {-1)" if r E n+1' n where n E N. Prove: f € R[0, 1], even though f has…

A: Sol

Q: t K = F(u), u algebraic over F and degree of u odd. Show that K = F(u²).

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Q: 7.2 Give an & - 8 pr at (1 continuous (2x - 2).

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Q: 4. Show, directly from Definition 7.33, that the function ƒ : R → R, f(x) = x² is analytic.

A: By definition, a real analytic function an open set U is one that can be expressed as a convergent…

Q: 4.1 Suppose that fig and g'are continuous on the interval 1= {x :asxs b). Show that a ) exists, and…

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Q: Show that the function f(z) = from [0, 0) to [0, ∞) has a fixed point c. Hint: Set f(r) = z and show…

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Q: 5.6.15. Let f: A → B be an arbitrary function. (a) Prove that if f is a bijection (and hence…

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Q: Sx, if xe Q if x ER – Q. f(x) = Using only the definition of derivatives and Lemma 4.2.2, determine…

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Q: 1. ( Prove that for any x, y e R, we have ||| - ||y|| < |× -y|l. 2. Let u e R. Prove that 0 u 0,…

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Q: Theorem 5-1. p (x;) = P (X = x;) = F (x) – F (x-1), where F is the d.f. of X.

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Q: Let f(x,y,z) = y² + 4z² – 2.ry – 2xz – 4yz + 1 be a function on the domain D where D = {(x,y,z) E R³…

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Q: 5. Find the x-coordinates for all points of discontinuity for f(x)- 「X316, xśl x〉 1 2 c. 1, 32d.…

A: First we check the limits as x approaches to 1 from left side and right side .

Q: 3.13. Let V D(0. 1). Suppose that f e C(V) NH(V), f(0) = 0 and |f(2)| < 1 for all z e V. Show that…

A: Given V=D(0,1) which is open disk with radius less than one. which is z&lt;1 Since f∈C(V¯)∩H(V)…

Q: 1. Show that if f: [a,b] → R and a < xo <b, then f'(xo) exists if and only if D f(x) = Dƒ(x) € R.

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Q: 5. Let f: Z→ Z be defined by f(x) = 2x-1. Determine whether f(x) is onto and/or 1 to 1. Explain your…

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Q: Determine whether or not f is a function. Justify your answers. 1 a) f: RR where f(x) for every XE R…

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Q: 10.50. Show that the function f : R → R defined by f (x) = 4x – 3 is bijective and determine f-1 (x)…

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Q: 5. Suppose that f: R→ R is continuous. and f(x) > 0.. Prove that there is an in- terval I =…

A: Used the definition of continuity.