Let f and g be functions defined on (possibly different) closed intervals, and assume the range of f is contained in the domain of g so that the composition g ◦ f is properly defined. (a) Show, by example, that it is not the case that if f and g are integrable, then g ◦ f is integrable. Now decide on the validity of each of the following conjectures, supplying a proof or  ounterexample as appropriate. (b) If f is increasing and g is integrable, then g ◦ f is integrable. (c) If f is integrable and g is increasing, then g ◦ f is integrable.

Q: c1x1(a) + c2x2(a)+ . . . +cnxn(a) = x(a). Let X (t) be the n x n matrix with column vectors x1, x2,…

A: Given : Xt be any  matrix of n×n           : c be column with component c1 ,c2 , . . .  . . .  ,cn…

Q: This exercise comes from Stein & Shakarchi, ch 2 # 6. Let be an open subset of C, and let TC be a…

A: We use Cauchy-Gaoursat Theorem.

Q: . If f is one-to-one and continuous on [a, b], prove that f must be strictly monotonic on [a, b].

A: It is given that f is one-to-one and continuous on [a, b] . then to prove that f must be strictly…

Q: (a) If ƒ and g are integrable, then gof is integrable. (b) If ƒ is increasing and g is integrable,…

A:

Q: Let f(z) be an entire function such that |f(z)| < 1 on the unit circle and f(a) = 0 for z - a some a…

A: Here given f(z)≤1 and we have to prove f(z)≤z-α1-α¯z Where z lies on unit circle and f(α)=0 for some…

Q: Let ƒ : R → R be a function. Give precise mathematical formulations of the following statements…

A:

Q: Let f: R -> R and E ⊆ R. Prove f is continous iff f-1(E) is closed whenever E ⊆ R is closed

A:

Q: Which Theorem states: If f is continuous function on the open interval (a, b) then f attains an…

A: Since the extreme value theorem states that- If f is a continuous function on [a,b] , then f has…

Q: Suppose that the function f is defined on a measurable set E and has the property that {x ∈ E|f (x)…

A:

Q: ): Discuss the continuity of the function is : x² – 1 X 2 f(x) = 2х — 1 - x+1 at x = 2.

A:

Q: Define f:R → R by f (x) = 5x if x is rational and f (x) = x² + 6 if x is irrational. Prove that f is…

A: Given, f:R→R by f(x)=5x if f(x) is rational and f(x)=x2+6 if x is irrational.To prove that f is…

Q: Suppose that f is a real valued function on the set S, which is contained in R, and that c is an…

A: We can solve using the definition of isolated point and continuity

Q: Q.Prove or disprove the following: if f and g are measurable functions, f+2g* then IS meaurable…

A: Constant functions are measurable.

Q: Let f be a function defined on ]0, 1[ by : 0, if x is irrațional f (x) = {1/q, if x=p/q; where p and…

A:

Q: Consider a function f:C-C, defined by f (z) = 72? . Then which of the following is true? Check all…

A: f is surjective.

Q: Let f be a function defined on ]0, 1I[ by : 0, if x is irrational x= p/q; where p and q are positive…

A:

Q: Let I  be an interval and let f : I → R be a continuous, injective function. Show that f is…

A:

Q: Consider the following statements 1- The function f defined by f(x) = x³ is decreasing in R 2- If a…

A:

Q: Let the function f : R → R be continuous, and suppose that K ⊆ R is compact. (a) Prove that f(K) ⊆…

A:

Q: Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval.…

A:

Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

A:

Q: the angle of rotation for the matrix transformation 2 2

A: Given the matrix of transformation is  f(xy)=32-121232xy To find the angle of rotation.

Q: Prove: There exist functions f, g, h in F such that g o f = h o f, but g is not equal to h.

A: The objective is to prove there exist functions f, g, h in F such that g o f = h o f, but g is not…

Q: True or false: If f: (0, ∞) → (0, ∞) is uniformly continuous, and h: (0, ∞) → (0, ∞) is defined by…

A:

Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

A:

Q: Show that if f and g are continuous on [a, b] and if f(a) g (b), then there exists some ce ]a, b[…

A:

Q: non-constant entire function such that |f| = 1 on the unit circle and f(1) = 1. Let f be a Show that…

A: Let f be a function as defined in the question. Define a new function via "double-conjugation…

Q: Q4. Simplify the following expression: 3-5 x 39 Click on image to enlarge if needed 12 81 243 324

A: To evaluate :- 3-5 * 39

Q: Se" –1 r >0 h(x) = |4- e" -1 x20 f(x) = sin x and g(x) = and x < 0 x <0

A:

Q: Suppose that f is holomorphic in an open set 2. Prove that in any one of the following cases: 2 3. f…

A:

Q: Prove or disprove: The integral of a bounded and measurable function over a set of finite measure is…

A: Proof: Suppose f be a bounded and measurable function over E, which is a set of finite measure. We…

Q: Suppose that f is a nonconstant continuous function defined on an interval I. a) Prove that f(I)…

A: Given that f is a non-constant continuous function. Then there exists x and y in the interval I such…

Q: Suppose f is entire and f(2)| s |2/0.9 for all z E C. Prove thatf is identically 0. (Hint: Follow…

A:

Q: Let ƒ : R → R be a continuous function such that f(0) 0. Prove that f(1)< 1 for all real x.

A: given f:ℝ→ℝ is a continuous function such that f(0)≤1 for all real x f(x)2-3f(x)+2≥0 to prove f(1)≤1…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.