Epsilon–Delta Continuity Quiz

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| Questions: 15 | Updated: Nov 24, 2025
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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.

Explanation

This is the formal ε–δ definition of continuity.

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About This Quiz
Epsilondelta Continuity Quiz - Quiz

Ready to practice the formal definition of continuity? This quiz helps you understand how ε–δ arguments connect input closeness to output closeness. You'll test your knowledge of when functions satisfy the ε–δ condition, how to construct δ for linear functions, and how limits relate to continuity. Through these questions, you’ll... see morebuild confidence in identifying whether a function is continuous at a point and how to structure an ε–δ proof correctly. By the end, you’ll have a strong foundation for handling formal limit and continuity arguments! see less

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

Explanation

Only continuous functions satisfy this condition—NOT every function.

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

Explanation

Continuity requires limit from both sides equals the function value.

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

Explanation

Differentiability implies continuity.

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

Explanation

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

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

Explanation

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

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

Explanation

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

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8) 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.

Explanation

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

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

Explanation

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

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

Explanation

δ measures input closeness; ε measures output closeness.

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

Explanation

Continuity at a means the limit equals the function value.

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

Explanation

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

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13) Which situation guarantees continuity at x = a?

Explanation

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

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

Explanation

|5x − 5a| = 5|x − a|.

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

Explanation

To make 2|x − a|

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If a function is continuous at a, then for every ɛ>0, there exists...
If |x−a| < δ, then automatically |f(x)−f(a)| < ɛ for any...
A function is continuous at a point if the left-hand limit, right-hand...
If a function is differentiable at a point, it must be continuous...
If a function is continuous at a point, then it must be differentiable...
If f is constant, then it is continuous everywhere.
If a limit does not exist at a point, the function cannot be...
To prove a function is not continuous at a point, it is enough to find...
In an ε–δ proof, δ is always equal to ε.
The ε–δ definition of limit deals with input closeness (δ)...
The statement “f is continuous at a” means:
In the ε–δ definition of continuity at a point a:
Which situation guarantees continuity at x = a?
If f(x) = 5x, then |f(x) − f(a)| =
For f(x) = 2x + 3, a suitable δ for a given ε is:
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