Consider the function: { x > 0 f(x) = %3D x +3 x < 0 a. Use a 8, e argument to prove that f(x) is discontinuous at x = 0. b. Find an open set U C R such that f-1(U) is not open and hence f(x) is not continuous on R.

Q: Let h: R → R be a continuous function on all of R. If h(x) Tt, prove that h(t) 50.

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A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

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

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A: I have proved limiting value is equal to functional value

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A: Recall: Properties of integral:  i)   ∫abf(x)dx=-∫baf(x)dxii) ∫abf(x)dx+∫bcf(x)dx=∫acf(x)dxiii)…

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A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

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Q: If f is continuous on [0, 1], prove that f₁¹ f(x) dx = f₁¹ ƒ(1 − x)dx.

A: This question can be solved using properties of definite integrals.

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Q: Every continuous function on R satisfies the conditions of MVT. * False True

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A: solution of above question is given below

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Q: Use the e-d definition to prove that f(x) = x³ + 3 is continuous at xo = 3.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.