Suppose f : [a, b] → R is is continuous and such that fg = 0 for all continuous functions g : [a, b] –→ R. What can you say about f? Justify your response in detail.

Q: Let f : R → R be a function with the following property: |f (a) – f(b)| < 4|a – b| for all a, b E R.

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Q: Modify the graphs of the functions ƒ and g as shown to obtain functions that are continuous on 3a,…

A: Given that the graph of functions y=fx and y=gx are given. We have to modify the graphs of the…

Q: Evaluate the function shown, where ƒ is continuous on [3, 8], ƒ(3) = 4,and ƒ(8) = 20.

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Q: Find numbers a and b so that f is continuous at every point.

A: Given,

Q: Consider a function f whose domain is the interval [a, b]. Show that if |f(x) − f(y)| ≤ (x − y)²,…

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Q: Determine the range of f, if f: D → R, where f(x) = x². I. D = (-2,2) II. D = [-7,2] III. D = (-4,2]…

A: I. D = (-2,2) Here the variable x lies between -2 and 2, that is,…

Q: let f be the function x²-C for x< 5 flx)=3x+5c for X Z 5

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Q: Formulate a condition for f to be symmetric with respect to the point (a, 0) on the x-axis.

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Q: Provide an appropriate response. 10x 7) Find an equation for the tangent to the curve y = at the…

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Q: Let f be the function from {a, b, c, d, e} to {1,2,3,4} with ƒ(a) = 3, ƒ (b) = 1, f(c) = 2, f(d) = 4…

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Q: Is the following statement correct? The continuity of the function f does not depend on the choice…

A: Continuity of function

Q: 3. Let f : A → B. Then 18f = f and f1a = f. Hint. You may use the fact that fg, for functions f, g,…

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Q: lim 3i 4 +

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Q: XI. Refer to the functions f, g, and h defined here. f(x) = 1/x, g(x) = 2 vx, and h(x) = x2 -1…

A: Using rules of functions.

Q: 4х if х> -1 f(x) 3D { 5 — х2 if b < x<-1 Зх + 7 if x < b

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Q: What values must a and b take for f to be continuous everywhere?

A: Given that the function fx=ax1+4x2 ,      x&lt;0b    ,                     x=0tan-1xx ,…

Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

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Q: Determine the domain and range of the function f: {r,s,t,u} ➔ {A,B,C,D,E} defined by f(r) = A, f(s)…

A: Domain of a function is the set of possible values of independent variable. The domain of given…

Q: 4. Definition: A set X CR is said to be disconnected if there exists disjoint open sets U and V such…

A: We have f: ℝ→ℝ, and X⊆ℝ is a connected set. We have to show that f(X) is connected.To prove this, we…

Q: Suppose f: R → R is a function. True or false: The composite function f of is continuous if and only…

A: True. Statement: Composition of f and f is continuous if and only if f is continuous. First, assume…

Q: Consider the function f : A → B.

A: Given the function f : A→ B. The relation  f^ from ℙ(A) to ℙ(B) is defined by…

Q: o f(x) f(x) is not a function because when x < 0 f(0) is undefined < 0 is undefined is defined

A: We have to complete the sentence.

Q: Let f : A → B. Then 1âf =ƒ and ƒ1a= f. Hint. You may use the fact that fg, for functions f, g, means…

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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.