Question

In: Advanced Math

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

Solutions

Expert Solution

a) Let f(x) be continuous at any point c.

Then there exists a sequence, converging to , such that the sequence converges to .

Then by definition, for , such that   for

for and ((By sequential criteria for continuity.))

Now by triangle inequality we know that :

for

Hence for a sequence converging to , the sequence converges to .

Therefore, by sequential criteria for continuity, is continuous.

b) Consider the function

It can easily be proven that this function is not continuous at any point.

But at every point. And is hence, continuous at each and every point.

Therefore, the converse may not be true. That is, if is continuous at a point , that does not mean that will be continuous at .


Related Solutions

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...
9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is...
9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]
a) State the definition that a function f(x) is continuous at x = a. b) Let...
a) State the definition that a function f(x) is continuous at x = a. b) Let f(x) = ax^2 + b if 0 < x ≤ 2 18/x+1 if x > 2 If f(1) = 3, determine the values of a and b for which f(x) is continuous for all x > 0.
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 ).
prove by epsilon-delta definition that f:R->R given by f(x)=x^3 is continuous at x=2
prove by epsilon-delta definition that f:R->R given by f(x)=x^3 is continuous at x=2
(a) State the definition of the derivative of f. (b) Using (a), prove the following:d/dx(f(x) +g(x))...
(a) State the definition of the derivative of f. (b) Using (a), prove the following:d/dx(f(x) +g(x)) =d/dx(f(x)) +d/dx(g(x))
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.
Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and...
Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and such that f(a)=f(b). Then there exists at least one point c on (a,b) such that f'(c)=0." Rolle's Theorem requires three conditions be satisified. (a) What are these three conditions? (b) Find three functions that satisfy exactly two of these three conditions, but for which the conclusion of Rolle's theorem does not follow, i.e., there is no point c in (a,b) such that f'(c)=0. Each...
Differential Geometry Open & Closed Sets, Continuity Prove f(t)=(x(t),y(t)) is continuous iff x(t) and y(t) are...
Differential Geometry Open & Closed Sets, Continuity Prove f(t)=(x(t),y(t)) is continuous iff x(t) and y(t) are continuous
prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) -...
prove using the definition of derivative that if f(x) and g(x) is differentiable than (f'(x)g(x) - f(x)g'(x))/g^2(x)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT