Epsilon–Delta Continuity Quiz

1) If a function is continuous at a, then for every ɛ>0, there exists a δ>0 such that whenever |x−a|<δ, we have |f(x)−f(a)|<ɛ.

True

False

This is the formal ε–δ definition of continuity.

Explanation

This is the formal ε–δ definition of continuity.

2) A function is continuous at a point if the left-hand limit, right-hand limit, and function value all agree.

True

False

Continuity requires limit from both sides equals the function value.

Explanation

Continuity requires limit from both sides equals the function value.

3) If a function is differentiable at a point, it must be continuous there.

True

False

Differentiability implies continuity.

Explanation

Differentiability implies continuity.

4) If a function is continuous at a point, then it must be differentiable there.

True

False

Continuity does not imply differentiability (example: |x| at 0).

Explanation

Continuity does not imply differentiability (example: |x| at 0).

5) If f is constant, then it is continuous everywhere.

True

False

A constant function never changes, so output closeness is automatic.

Explanation

A constant function never changes, so output closeness is automatic.

6) If a limit does not exist at a point, the function cannot be continuous there.

True

False

Continuity requires the limit to exist and equal f(a).

Explanation

Continuity requires the limit to exist and equal f(a).

7) To prove a function is not continuous at a point, it is enough to find one ε such that no δ satisfies the ε–δ condition.

True

False

Showing the ε–δ condition fails for even one ε proves discontinuity.

Explanation

Showing the ε–δ condition fails for even one ε proves discontinuity.

8) The ε–δ definition of limit deals with input closeness (δ) producing output closeness (ε).

True

False

δ measures input closeness; ε measures output closeness.

Explanation

δ measures input closeness; ε measures output closeness.

9) In the ε–δ definition of continuity at a point a:

ε controls input closeness

δ controls output closeness

ε controls output closeness

δ = ε always

ε controls output closeness; δ ensures inputs are close enough.

Explanation

ε controls output closeness; δ ensures inputs are close enough.

10) Which situation guarantees continuity at x = a?

Both one-sided limits exist and equal f(a)

Only left-hand limit exists

Only right-hand limit exists

F(a) is undefined

Continuity requires both one-sided limits to equal f(a).

Explanation

Continuity requires both one-sided limits to equal f(a).

11) If f(x) = 5x, then |f(x) − f(a)| =

|x − a|

5|x − a|

10|x − a|

|5 − a|

|5x − 5a| = 5|x − a|.

Explanation

|5x − 5a| = 5|x − a|.

12) For f(x) = 2x + 3, a suitable δ for a given ε is:

δ = ε

δ = 2ε

δ = ε/2

δ = 3ε

To make 2|x − a|

Explanation

To make 2|x − a|

13) In an ε–δ proof, δ is always equal to ε.

True

False

δ depends on the function. Sometimes δ = ε, but often δ = ε/(something).

Explanation

δ depends on the function. Sometimes δ = ε, but often δ = ε/(something).

14) If |x−a| < δ, then automatically |f(x)−f(a)| < ɛ for any function.

True

False

Only continuous functions satisfy this condition—NOT every function.

Explanation

Only continuous functions satisfy this condition—NOT every function.

15) The statement “f is continuous at a” means:

F(a) exists

Limₓ→ₐ f(x) = f(a)

Limₓ→ₐ f(x) does not exist

F must be differentiable

Continuity at a means the limit equals the function value.

Explanation

Continuity at a means the limit equals the function value.

Epsilon–Delta Continuity Quiz

1) If a function is continuous at a, then for every ɛ>0, there exists a δ>0 such that whenever |x−a|<δ, we have |f(x)−f(a)|<ɛ.

2)

What first name or nickname would you like us to use?

2) A function is continuous at a point if the left-hand limit, right-hand limit, and function value all agree.

3) If a function is differentiable at a point, it must be continuous there.

4) If a function is continuous at a point, then it must be differentiable there.

5) If f is constant, then it is continuous everywhere.

6) If a limit does not exist at a point, the function cannot be continuous there.

7) To prove a function is not continuous at a point, it is enough to find one ε such that no δ satisfies the ε–δ condition.

8) The ε–δ definition of limit deals with input closeness (δ) producing output closeness (ε).

9) In the ε–δ definition of continuity at a point a:

10) Which situation guarantees continuity at x = a?

11) If f(x) = 5x, then |f(x) − f(a)| =

12) For f(x) = 2x + 3, a suitable δ for a given ε is:

13) In an ε–δ proof, δ is always equal to ε.

14) If |x−a| < δ, then automatically |f(x)−f(a)| < ɛ for any function.

15) The statement “f is continuous at a” means: