1. Use the ε-δ definition of continuity to prove that (a) f(x) = x 2 is...

1. Use the ε-δ definition of continuity to prove that (a) f(x) = x 2 is continuous at every x0. (b) f(x) = 1/x is continuous at every x0 not equal to 0.

3. Let f(x) = ( x, x ∈ Q 0, x /∈ Q (a) Prove that f is discontinuous at every x0 not equal to 0. (b) Is f continuous at x0 = 0 ? Give an answer and then prove it.

4. Let f and g be two functions on R. Prove that if f and g are continuous on R and f(q) = g(q) for every q ∈ Q, then f(x) = g(x) for every x ∈ R.

Expert Solution

Colby Messinger answered 2 years ago

The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for...

The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for every ε > 0 there is a δ > 0 such that for any x with |x − a| < δ we are guaranteed to have |f(x) − L| < ε as well. 1. Use the ε − δ definition of limits to verify that limx→1 (−2x + 1) = −1. [2] 2. Use the definition of limits that you didn’t use in answering...

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous. b) theorem 17.3...

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous. b) theorem 17.3 states that if f is continuous at x0, then |f| is continuous at x0. is the converse true? if so, prove it. if not find a counterexample. hint: use counterexample

Use the ε-δ definition of limits to prove that lim x3 −2x2 −2x−3 = −6. 3...

Use the ε-δ definition of limits to prove that lim x3 −2x2 −2x−3 = −6. 3 markx→1 Hint: This question needs students have a thorough understanding of the proof by the ε-δdefinition as well as some good knowledge of what is learnt in Math187/188 and in high school, such as long division, factorization, inequality and algebraic manipulations. End of questions.

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x...

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x + 8 is convex. Is this function strictly convex? 2. Use the definition of convexity to prove that the function f(x)= ax + b is both convex and concave for any a•b ≠ 0.

. Let f(x) = 3x^2 + 5x. Using the limit definition of derivative prove that f...

. Let f(x) = 3x^2 + 5x. Using the limit definition of derivative prove that f '(x) = 6x + 5 Then, Find the tangent line of f(x) at x = 3 Finally, Find the average rate of change between x = −1 and x = 2

2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for...

2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for all xin an interval ( a , b ), then f is constant on ( a , b ). b True or False. The product of two increasing functions is increasing. Clarify your answer. c Find the point on the graph of f ( x ) = 4 − x 2 that is closest to the point ( 0 , 1 ).

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1. Use the ε-δ definition of continuity to prove that (a) f(x) = x 2 is...

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The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for...

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