Let f:(1,3)→ R be continuous on (1, 3). Then, for each & >0 and each u € (1,3) there exists a 5(e,u) >0 such that |f(x)-f(u)| <ɛ whenever |x-u|

Q: Suppose that the function f:(a, b) → R is uniformly continuous. Prove that f: (a, b) → Ris bounded.

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Q: Theorem 7. Let X be a continuous RV, g a function from R to R. Let Y = g(X). Then E[Y] = E[g(X)] = /…

A: The expected value of random variable is the weighted average of all possible values of the…

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Q: Suppose that f(x) is a continuous function on [a , b] and F(x)= | f(t) dt a Part 1 of the…

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Q: Suppose a and b are continuous functions on [a, b] and that g(x) ≥ 0 for all x ∈ [a, b]. Prove that…

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Q: Let f(x) be a continuous function on R-{0} such that (f(x) dx exists for all a > 0.If g(x)= |- dt, t…

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,... then show that f(x)=0…

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Q: Prove that f(z)=|z| is nowhere differentiable in C but is continuous everywhere.

A: Given function is fz=z. We have to prove that the function fz is nowhere differentiable in C but is…

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Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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Q: 3. Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + Y (x, y) # (1, –1) x +y is…

A: Here, the given function is: f(x,y)=1;   (x,y)=(1,-1)x2+yx+y; (x,y)≠(1,-1) lim(x,y)→(1,-1)

Q: Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0<|e'(x)| ≤ |f'(x)| for all x ∈ (0,…

A: Given: ex and fx are differentiable on 0,1 and that 0&lt;e'x≤f'x, ∀x∈0,1. To Show: ex-ey≤fx-fy,…

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Q: Let f:X→R be a Lipschitz continuous function defined on X. Show that f is uniformly continuous on X.

A: Given : function f:X -&gt; R is a Lipschitz continuous function defined on X . To show : f is…

Q: Let if (x, y) # (0, 0) f(x, y) = { if (x, y) = (0, 0). (a) Compute the limit as (x, y) → (0, 0) of f…

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Q: Show that the function f defined by (x, y) = (1, –1) f(r, y) = { x2 + y (x, y) # (1,–1) r + y is not…

A: To show that the function f defined by fx, y=1, x, y=1, -1x2+yx+y, x, y≠1, -1is not continuous at 1,…

Q: Suppose that f: [0, ) – R is continuous, and that f(x) – l as x → - x. Prove that f is uniformly…

A: See the attachment

Q: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

Q: Prove that if f is continuous on [a, b] and F(x) = ſx ƒ (t)dt α then F is continuously…

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Q: Let f : [0 , 1] → R be a continuous function on [0 , 1] such that : f(x)x" dx = 0, (n = 0, 1, 2,…

A: Given that f:0,1→R be a continuous function on 0,1 such that ∫01f(x)xndx=0. The objective is to show…

Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

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Q: I. Show that the function f defined by 1, (x, y) = (1, –1) %3D f(r, y) = { r2 + y (x, y) + (1, –1)…

A: Given function is  fx,y=1,             x,y=1,-1x2+yx+y,    x,y≠1,-1

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,.. then show that f(x)3D0…

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Q: Suppose that f:[0, 1] → R is continuous on [0, 1] and differentiable on (0, 1) with f (0) = 0 and…

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Q: Let Y be any topological space and y: [0, 1] → Y be a continuous function so that r(0) = r(1) = xp.…

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Q: Show that the function f defined by (x, y) = (1, – 1) f (x, y) : x2 + Y (x, y) + (1, – 1) x + y is…

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Q: Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e"…

A: Differentiate the equation wrt x:  ddxyx=ddxex+yx22y'x=ddxex+ddxyx22y'x=ex+xy'x2

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A: Let F be defined on RXR→R by F(x,y) =x2-y. Now we have to show that F is in class C' on a…

Q: Show that the function f defined by 1, (x, y) = (1, – 1) f (x, y) : .2 x + y (x, y) # (1, – 1) x + y…

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Q: If f is a continuous function on [a, b], ƒ is differentiable on (a, b), and if both f(a) and f(b)…

A: This question is based on Rolle's theorem which is explained in step 2.

Q: Let f(x) = 2x+1, if x1 O (-∞, 1) U (1,00) O (-∞0, 1] U (1, ∞0) OR O (-∞0, 1) U [1,00)

A: topic- limits and continuity

Q: ) Prove that the envelope of the family of f (x, y, a) = 0 member of the family at a characteristic…

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Q: Let f : [0, 1] → [0, 1] be continuous. Show that there exists an element x = [0, 1] such that f(x) =…

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Q: Prove that if f is a real-valued continuous function on [a, b] × [c, d], then f takes on a maximum…

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Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y…

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Q: Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) f2(1).…

A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .

Q: Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all r,y R. Prove that for some…

A: What is Continuous Function: In mathematics, a continuous function is one that, in response to a…

Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

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Q: Let f: X→Y, g:Y-Z be uniformly continuous. Then gof:X-Z is uniformly continuous.

A: The main objective is to proof that the function g∘f is uniformly continuous, i.e. for every ε&gt;0,…

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = (x, y) = (0,0) y+2xy² x²+y2 Let…

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Q: Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] and on [1,∞). Show that it is…

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Q: Assume that J is continuous on [0, 3 and differentiable on (0, 3) with f (z) > 6 for each z in (0,…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

A: Given function is fx,y=1,           x,y=1,−1x2+yx+y  ,x,y≠1,−1 . We have to show that the given…

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Q: Show that f(x) = 1/x2 is uniformly continuous on the set [1,∞) but not on the set (0, 1].

A: To show that f(x)=1x2 is uniformly continuous on the set [1,∞) but not on (0,1].

Q: (5) Suppose that fis continuous on [a, b]- f(x) ≥ 0 Vx = [a,b] and [f(x)dx = 0. Prove that f(x) = 0…

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Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.