Suppose a function a increases on [a, b}, a xoSb, a is continuous at xp. Let f be a function defined in [a,b] such that f (xo) = 1 and %3D b f (x) = 0 for x# xo. Prove that f ɛR (a) on [a, b] and f da= 0. a

Q: Suppose a function a increases on [a, b}, a < xoS b, a is continuous at xo. Let f be a function…

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A: Topic- Limits

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A: Let us define function f : R ---&gt; R by f(x) = x2 Clearly f(x) is continuous  We know that, The…

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A: Explained below

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Q: Suppose a function a increases on [a, b}, a < xo S b, a is continuous ut xo. Let f be a function…

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Q: iii. is continuous at a provided g(a) # 0.

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A: Find we have to use definition of derivative.

Q: Suppose a function a increases on [a, b}, a S xo Sb, a is continuous at xg. Let f be a function…

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A: A function is strictly increasing if for all x1 , x2 such that x1&lt;x2 , we have  fx1&lt;fx2. Now…

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Q: Suppose f ond 9 are continuous functions on To.b] ond thead gel >0 for all x E [o,b]. Prove that…

A: Using the Concept of Riemann Integration to prove the Given Theorem

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A: Consider the given information.

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Q: Suppose f is a continuous function defined on (-e, e) for some e > 0 and assume that f(0) = 0 and…

A: Since function f(x) is continuous at x=0. Also given that limx→0f(x)x=C

Q: Suppose f is continuous on a closed interval [a, b]. If F(x) is an antiderivative of f(x), for real…

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Q: Isg continuous on (0,1]? * if x 4.

A: Solution:Given  g(x)=xif x≤03if 0&lt;x≤13-x2if 1&lt;x≤4x-3if x&gt;4To find:g is continuous at (0,1]

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Q: Consider the function f defined on [0,∞) f(x) = xrsin(1/x), for x≠0 and f(x) = 0, for x=0 where r >…

A: Given: f(x)=xr sin1x,   x≠00         ;     x=0     a) f is continuous on [0,∞) f is continuous if…

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Q: Suppose ƒ is continuous on a closed interval [a, b]. If F(x) is an antiderivative of f(x),for real…

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Q: ) Let f be a continuous (and thus uniformly continuous) function on [a, b]. Show that, given e > 0,…

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

Q: Suppose a function a increases on [a, b}, a S xo S b, a is continuous at xo. Let f be a function…

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Q: Suppose a function a increases on [a, b}, a < xo Sb, a is continuous at xo. Let f be a function…

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Q: (a) Let f be differentiable on (a, b) and suppose that f'(t)| 0 and for all t e (a, b). Prove that…

A: We solve here 3(a), In question given two problem are separated problem. According to our guideline,…

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Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and if…

A: detailed explanation mentioned below.

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A: Here given function is continuous and strictly increasing function on [0,1]

Q: Suppose f is continuous and F'(x) = f(x). J s(x)dx =3, f(x)dx = 4, [f(x)dx= 5, and F(1) =6, evaluate…

A: Here this is simple application of integration and differentiation please watch carefully.

Q: Suppose f is continuous on the closed interval [a.b]. if F is an antiderivative of f for all real…

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