Discontinuity Types And Behavior Quiz

1) Which of the functions has a removable discontinuity at x = 2?

F(x) = x^2 - 4

F(x) = 1/(x-2)

F(x) = ⌊x⌋

F(x) = tan(x)

(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.

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.

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

Removable discontinuity at x=0

Jump discontinuity at x=0

Infinite discontinuity at x=0

Continuous everywhere

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

Explanation

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

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

Removable

Jump

Infinite

Oscillatory

Limit exists but function value wrong/missing → removable.

Explanation

Limit exists but function value wrong/missing → removable.

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

E^x

Sin(x)

Sin(x)/x

Tan(x)

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

Explanation

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

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

Left and right limits equal

Left and right limits differ

Limit equals f(a)

F(a) undefined

Jump requires unequal one-sided limits.

Explanation

Jump requires unequal one-sided limits.

6) Which is an example of infinite discontinuity?

1/x^2 at x=0

|x|

X^2

⌊x⌋

1/x^2 → ∞ as x → 0.

Explanation

1/x^2 → ∞ as x → 0.

7) The function has (typical exam reference):

Removable discontinuity at x=1

Jump discontinuity at x=1

Infinite discontinuity at x=1

Continuous at x=1

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

Explanation

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

8) Which is not a type of discontinuity?

Jump

Removable

Infinite

Continuous

Continuous is not a discontinuity type.

Explanation

Continuous is not a discontinuity type.

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

0

π

1

∞

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

Explanation

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

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

X^2

Dirichlet function

E^x

⌊x⌋

Dirichlet function is discontinuous everywhere.

Explanation

Dirichlet function is discontinuous everywhere.

11) Discontinuous functions cannot be integrated.

True

False

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

Explanation

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

12) All discontinuities are removable.

True

False

Only removable discontinuities can be fixed.

Explanation

Only removable discontinuities can be fixed.

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

True

False

Limit as x → 0 does not exist.

Explanation

Limit as x → 0 does not exist.

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

True

False

Continuous + continuous = continuous; cannot create jumps.

Explanation

Continuous + continuous = continuous; cannot create jumps.

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

True

False

Vertical asymptote means limit → ∞ → discontinuous.

Explanation

Vertical asymptote means limit → ∞ → discontinuous.

Discontinuity Types and Behavior Quiz

1) Which of the functions has a removable discontinuity at x = 2?

2)

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

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

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

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

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

6) Which is an example of infinite discontinuity?

7) The function has (typical exam reference):

8) Which is not a type of discontinuity?

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

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

11) Discontinuous functions cannot be integrated.

12) All discontinuities are removable.

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

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

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