Let f : [a, b] –→ R be a continuous function and suppose that N is a natural number. If 11. ..., XN are real numbers [a, b), prove that there is some xo in [a, b] such that f(r1) + ..+ f(xN) f(ro) = N

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

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Q: Prove that f(x) = |x| is continuous at all points c in ℝ.

A: Continuity of a function can be done by many methods as,

Q: Let f : R² → R defined by f((x1, x2)) = x1 - x2. Then determine if is Lipschitz continuous(if it is,…

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Q: Suppose f:[0, 27] → R is continuous and f(0) = f(2n). Prove that there exists at least one point c E…

A: This question is based on intermediate value theorem .  The theorem says ,  If f is any continuous…

Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

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Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

Q: 4. Assume that a continuous function f : R –→ R is T-periodic; that is, f(x + T) = f(x) for all x E…

A: given a continuous function f:ℝ→ℝ is T-periodic, defined as fx+T=fx ∀x∈ℝ to prove- for every natural…

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Q: Let f : R → R, and let x E R. Prove that f is continuous at x if and only if for every e > 0 there…

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Q: Prove that if f: A to R is continuous at x=c, then for all (x_n) converging to c, it follows that…

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Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

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Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

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Q: D. Prove that f(x) = |x - 6| is continuous at all points c in R.

A: Given function is fx=x−6. We have to prove that the function fx=x−6 is continuous at all point c in…

Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

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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: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

Q: Find all values of "c" such that the function f(x) is continuous over  ( − ∞ , ∞ ) f(x) = 1 - x^2…

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Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

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Q: . If f is a continuous function from the metric space X to the metric space Y and {xn} is a sequence…

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Q: Let f : [a, b] → R be a continuous function and suppose that N is a natural number. If ¤1,..., IN…

A: It's a question from intermediate value theorem, so let's see how we can do it.

Q: Let f be a continuous function on R such that limr→∞ f (x) = L* and lim,-0 f(x) = L-. Assume L+ > L¯…

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Q: • Assume f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is…

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Q: f (x)dx = f(c)(b – a). a

A: given: the function f is continuous on the closed interval [a, b], then there exists at least one…

Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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Q: Let f and g be continuous functions on [a, b], and suppose that f(a) g(b). Prove that f(c) = alc)…

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Q: If f(x) is differentiable in (a, b) and there is M > 0 such that |f'(x)| < M for all x E (a, b),…

A: Topic = Functions True

Q: Prove that if f is continuous in [a, b] and c is the infimum of f in [a, b], then there exists an xo…

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Q: f(c) = g(c).

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Q: Prove that if S(c +h) – f(c) lim h exists, then f is continuous at c.

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Q: If f, g : R R s.t. f and g are uniformly continuous. and bounded, then show that f g is also…

A: A differentiable function f:D→R is said to be uniformly continuous on D if for all ∈&gt;0 , ∃ δ&gt;0…

Q: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

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Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: Find all values of a such that f is continuous on R: |x + 1 if x a

A: For a continuous function LHL ( left hand limt) should be equal to RHL (right hand limit)

Q: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

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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…

Q: If f is a continuous function on [a, b], and if f (a) = -2 and ƒ (b) = 2, then there exists a real…

A: Hi, you can find the answer in the handwritten form.  Given f(a)=-2  f(b)= 2

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Q: Show (in terms of  −δ) that a function f : [2,7] −→R be defined by f(x) = √x2+1 is uniformly…

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Q: Prove that if f is a continuous function with domain [0, 1] and X is the set of all x in [0, 1] such…

A: F is a continuous function with domain (0,1) and X is the set of all x in (0,1) such that F(x) = 0.…

Q: Let f be a continuous function on [a, b] such that f(r)a" dx = 0 for every n > 0. Prove that f = 0…

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Q: x + 6|, if x - 1 x² – 3x

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Q: Let f:R" → R be a continuous function, and let x1, x2 € R" be distinct. Prove that there exist a…

A: Given: Now consider the given, Let f:ℝn→ℝ be a continuous function, and x1, x2 ∈ℝn  be two…

Q: Suppose f: (a, b) → (a, b| is a continuous function. Prove that there exists ace Ja, b) such that C.

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Q: If f(x) is continuous on a closed interval I, and f(a) and f (b) have opposite signs where a and b…

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Q: Suppose the function f has the property that there exists a number B such that |S(x) – f(c)| < B|x –…

A: Suppose the function f has the property that there exists a number B such that |f(x)-f(c)|≤B|x – c|…

Q: Assume that is a continuous real function on (a, b) such that 1 1 o ( x + ³) ≤ ²}/20 (x) + = 0 ( for…

A: Here given that φ is a continuous real function on a, b such that φx+y2≤12φx+12φx

Q: Prove that if f is a continuous function with domain 0, 1] and X is the set of all x in 0, 1] such…

A: Given that f is a continuous function with domain 0,1. Let, X=x| x∈0,1 and fx=0. Also, given that s…

Q: If f is a continuous function on R with (1) > 0 and f(4) < 0, then there exists a number c between 1…

A: In this question, we need to check whether the given function is true or false.

Q: Assume f : [a, b] → R is a continuous function on the closed interval [a, b] and differentiable on…

A: Assume f : [a, b] → R is a continuous function on the closed interval [a, b] and differentiable on…

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.