Discontinuity Types and Behavior Quiz

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| Questions: 15 | Updated: Nov 24, 2025
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1) Which of the functions has a removable discontinuity at x = 2?

Explanation

(x^2 - 4) factors as (x-2)(x+2). If viewed as (x^2-4)/(x-2), it simplifies to x+2 but is undefined at x=2 → removable.

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About This Quiz
Discontinuity Types And Behavior Quiz - Quiz

Think you can spot discontinuities quickly? This quiz challenges you to identify when a function is discontinuous and what type of discontinuity it has. You’ll analyze behavior near undefined points, compare one-sided limits, and explore classic examples such as 1/x, floor functions, and oscillatory functions. Through these questions, you'll understand... see morethe differences between removable, jump, and infinite discontinuities—and see how piecewise and rational functions behave near problem points. By the end, you’ll feel confident classifying discontinuities and explaining why they occur! see less

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2) The function f(x) = 1/x has:

Explanation

1/x blows up to ±∞ near x=0 → infinite discontinuity.

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3) If lim_{x→a−} f(x) = lim_{x→a+} f(x) ≠ f(a), then the discontinuity is:

Explanation

Limit exists but function value wrong/missing → removable.

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4) Which of the following functions is discontinuous at x = 0?

Explanation

sin(x)/x is undefined at x=0 → removable discontinuity.

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5) A function has a jump at x = a if:

Explanation

Jump requires unequal one-sided limits.

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6) Which is an example of infinite discontinuity?

Explanation

1/x^2 → ∞ as x → 0.

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7) The function has (typical exam reference):

Explanation

Typical rational-simplification problem with a hole at x=1.

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8) Which is not a type of discontinuity?

Explanation

Continuous is not a discontinuity type.

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9) F(x) = sin(1/x) has a discontinuity at:

Explanation

sin(1/x) oscillates infinitely as x → 0 → no limit.

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10) Which function has discontinuities at infinitely many points in [0,1]?

Explanation

Dirichlet function is discontinuous everywhere.

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11) Discontinuous functions cannot be integrated.

Explanation

Many discontinuous functions (e.g., step functions) are integrable.

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12) All discontinuities are removable.

Explanation

Only removable discontinuities can be fixed.

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13) F(x) = sin(1/x) is continuous at x=0.

Explanation

Limit as x → 0 does not exist.

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14) Functions with jump discontinuities can be represented as sums of continuous functions.

Explanation

Continuous + continuous = continuous; cannot create jumps.

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15) If a function has a vertical asymptote at x = a, it is discontinuous at x = a.

Explanation

Vertical asymptote means limit → ∞ → discontinuous.

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Which of the functions has a removable discontinuity at x = 2?
The function f(x) = 1/x has:
If lim_{x→a−} f(x) = lim_{x→a+} f(x) ≠ f(a), then the...
Which of the following functions is discontinuous at x = 0?
A function has a jump at x = a if:
Which is an example of infinite discontinuity?
The function has (typical exam reference):
Which is not a type of discontinuity?
F(x) = sin(1/x) has a discontinuity at:
Which function has discontinuities at infinitely many points in [0,1]?
Discontinuous functions cannot be integrated.
All discontinuities are removable.
F(x) = sin(1/x) is continuous at x=0.
Functions with jump discontinuities can be represented as sums of...
If a function has a vertical asymptote at x = a, it is discontinuous...
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