f is continuous at a if and only if |f| is continuous at a.

Q: If f is differentiable at a , then f is continuous at a . True O False

A:

Q: Assume that a continuous function f:R → R is X- and µ-periodic, where A and u are two nonzero…

A: We know that between two rational number lies infinite many irrational number. But function is…

Q: If f is continuous at a number, x, then f is differentiable at x. True

A:

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: 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: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

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Q: Show that if f(x) is continuous in(a, b),then there exists a point in(a, b) such that…

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Q: If f is continuous at a, then f is differentiable at a. O True O False

A: Answer: False.

Q: Let f : E → R be continuous such that |f(x)|≤|x|, for all x ∈ E. Is f uniformly continuous, justify

A: According to the given information,

Q: Let f and g be continuous functions on the interval [a,b], such that f(x)=g(x)-1 for all asx<b.…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: c =

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Q: x< x3 B- If f(x) = Ax? is continuous at x = ; Find A? HIN IN

A: We are given : f(x)=x3       , x&lt;12Ax2     ,x≥12 is continuous at x=12 Condition for f(x) to be…

Q: If f(x) is continuous at x = c then f'(c) exists. True False

A:

Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: Prove that the function f(x) = 2x is uniformly continuous on R.

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Q: Use the definition of continuity to determine whether f is continuous at a.

A: Given :           f(x) =  1 - x      if x &lt; 1             0           if x = 1            x2 - 1…

Q: Show by definition that f(x) = |x| : R → R is continuous on R. 2.

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Q: Let f be continuous on [a, b] and suppose that f(x) 2 0 for all x e [a,b]. Prove that if L(f)=0,…

A: Given: Let f be continuous on [a,b] and suppose that f(x) is greater than or equal to zero for all x…

Q: Prove that f(x) =√x is uniformly continuous on [0,∞).

A: According to the question, we have to prove that the function fx=x is uniformly continuous on [0,∞).…

Q: Let f : E → R be uniformly continuous. Prove that |f| is uniformly continuous. Is the converse true,…

A: Let f: E → R be uniformly continuous.  Now, we need to show that |f| is uniformly continuous.  Now,…

Q: Let f be a continuous function defined on E. Is it true that f-'(A) is always measurable if A is…

A: If you mean AA is Lebesgue measurable, then the conclusion is NO. A classic counterexample is…

Q: If a function f(x) is not defined at x-c, then it is not continuous at a close interval [a, c). * O…

A: We know that function is continuous at c if the 1) f(c) is defined 2)limit of f(x) exist, Here…

Q: If a function f(x) is not defined at x-z, then it is not continuous at interval (a, b) where a > z <…

A: Solution

Q: Let g : R → R be a positive function such that g(0) = 1 and g(x+y) = g(x)g(y), ∀x, y ∈ R. Show that…

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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…

A:

Q: If a function f(x) is not defined at x-a, then it is not continuous at an open interval (a, b) * O…

A: Given Statement is "False"

Q: Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one real…

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Q: Determine if g (r)= is continuous at a = 8.

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Q: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

A: Given function is fx=x−1x+1. We have to show that fx=x−1x+1 is uniformly continuous on 0,∞. Let us…

Q: If a function f(x) is not defined at x-a, then it is not continuous at x-a.

A:

Q: (a) Prove that if m ₹ N, then the function f(x) = xm is continuous on R.

A: We have to prove that, the function f(x) = xm,   m∈ℕ  is continuous on ℝ.

Q: Where is f(a) not continuous?

A: Given:

Q: Let f(x) = 0 if x∈Q if x and f(x) = 1 if x ∈R\Q. Prove that y=f(x) is not continuous at any point…

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

A:

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

Q: Find c such that the function x ≤ c { is continuous everywhere. f(x) = x² - 3 X 6x 12 x > c -

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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: 5.Prove that: if f is continuous at x = 0, then xf (x) is differentiable at x = 0.

A: We want to prove the given result.

Q: 4. Let A := (0, 1] and let f : A → R be defined by f(x) = !. Prove that f is continuous on A.

A:

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: Suppose the function ƒ has the property that IS(x) – S()| < |x -|| for each pair of points x, t in…

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Q: Let f: (0,0) → (0, 0) be a function defined by f (x) =- for all x E (0, ∞). Prove that fis…

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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: Let f be a continuous function defined on E. Is it true that f^-1(A)  is always measurable if A is…

A: In this question we have to find if is a continuous function defined on E and Is that that true if…

Q: øz (@) is a continuous function of o in the range -o < @< ∞.

A:

Q: Prove: f(x)=5x y is continuous at x=2

A: A function fx is continuous at a point x=a, if limx→afx=fa Given function is

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].

Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.