Discontinuous Functions Quiz

1) A function can be discontinuous at more than one point.

True

False

A function may have multiple discontinuities (e.g., piecewise or rational functions).

Explanation

A function may have multiple discontinuities (e.g., piecewise or rational functions).

2) If a function is continuous on a closed interval [a,b], it cannot have any jump discontinuity in that interval.

True

False

Jump discontinuities violate continuity on the interval.

Explanation

Jump discontinuities violate continuity on the interval.

3) A removable discontinuity occurs when a function is undefined at a point, but the limit exists.

True

False

If the limit exists but the value is missing or wrong, the discontinuity is removable.

Explanation

If the limit exists but the value is missing or wrong, the discontinuity is removable.

4) A jump discontinuity occurs when the left-hand limit and right-hand limit at a point exist but are not equal.

True

False

A “jump” occurs when the two one-sided limits both exist but differ.

Explanation

A “jump” occurs when the two one-sided limits both exist but differ.

5) A function discontinuous at a point must have a limit that does not exist at that point.

True

False

Removable discontinuities have limits; the value is the issue.

Explanation

Removable discontinuities have limits; the value is the issue.

6) The Heaviside function is an example of a function with jump discontinuities.

True

False

The Heaviside step function jumps at 0.

Explanation

The Heaviside step function jumps at 0.

7) Removable discontinuities can be eliminated by redefining the function at the discontinuous point.

True

False

If the limit exists, redefining f(a)=limit removes the discontinuity.

Explanation

If the limit exists, redefining f(a)=limit removes the discontinuity.

8) A function with an infinite discontinuity has the property that its limit approaches infinity at some point.

True

False

Infinite discontinuity means the function blows up near a point.

Explanation

Infinite discontinuity means the function blows up near a point.

9) Piecewise-defined functions are always discontinuous.

True

False

Piecewise functions can be continuous (e.g., |x|).

Explanation

Piecewise functions can be continuous (e.g., |x|).

10) Functions with oscillatory behavior near a point can have a discontinuity.

True

False

Extreme oscillation (e.g., sin(1/x)) can destroy the limit.

Explanation

Extreme oscillation (e.g., sin(1/x)) can destroy the limit.

11) If limₓ→a f(x) exists but f(a) is undefined, the discontinuity is:

Removable

Jump

Infinite

Essential

Limit exists but value missing → removable.

Explanation

Limit exists but value missing → removable.

12) Which of the following statements is true?

Infinite discontinuities occur only when function tends to infinity

Jump discontinuities occur when limits are infinite

Removable discontinuities occur only when limits do not exist

All discontinuities are infinite

Infinite discontinuities happen when f(x) → ±∞.

Explanation

Infinite discontinuities happen when f(x) → ±∞.

13) F(x) = 1/(x² − 1) is discontinuous at:

X = 1

X = −1

X = 0

Both A and B

x² − 1 = 0 at x=±1 → discontinuities.

Explanation

x² − 1 = 0 at x=±1 → discontinuities.

14) If limₓ→0 f(x) = L but f(0) ≠ L, the discontinuity is:

Removable

Jump

Infinite

None of the above

Limit exists but function value is wrong → removable.

Explanation

Limit exists but function value is wrong → removable.

15) The function f(x) = (x² − 9)/(x − 3) has:

Removable discontinuity at x = 3

Jump discontinuity at x = 3

Infinite discontinuity at x = 3

Continuous at x = 3

Simplifies to x+3 but undefined at 3 → removable.

Explanation

Simplifies to x+3 but undefined at 3 → removable.