Fundamentals Of Continuity: Limits, One-Sided Limits, And Function Behavior

1) If a function is continuous at every point of its domain, it must also be continuous on any subset of its domain.

True

False

Any subset inherits continuity from the original domain.

Explanation

Any subset inherits continuity from the original domain.

2) If f is continuous on [a,b], then f must attain both a maximum and a minimum on that interval.

True

False

Extreme Value Theorem ensures max & min on closed intervals.

Explanation

Extreme Value Theorem ensures max & min on closed intervals.

3) If a function has a jump discontinuity, then the left-hand and right-hand limits both exist but are not equal.

True

False

Jump means left and right limits exist but differ.

Explanation

Jump means left and right limits exist but differ.

4) A function can be continuous at a point even if it is not defined in any open interval around that point.

True

False

Continuity only requires the limit equals the function value.

Explanation

Continuity only requires the limit equals the function value.

5) If f(x) is continuous and never zero, then 1/f(x) is also continuous.

True

False

Reciprocal of a nonzero continuous function is continuous.

Explanation

Reciprocal of a nonzero continuous function is continuous.

6) Which of the following is not necessarily continuous?

The sum of two continuous functions

The composition of two continuous functions

The maximum of two continuous functions

The inverse of a continuous function

Inverse need not be continuous unless function is bijective & monotone.

Explanation

Inverse need not be continuous unless function is bijective & monotone.

7) Let Is f continuous at x=0?

Yes

No

Function definition missing → cannot determine continuity.

Explanation

Function definition missing → cannot determine continuity.

8) Which function fails to be continuous at exactly one point?

F(x)=x^3

F(x)=|x|

F(x)=|x|/x

F(x)=e^x

Only |x|/x has a discontinuity at x=0.

Explanation

Only |x|/x has a discontinuity at x=0.

9) If f is continuous on [2,6], which of the following must be true?

F(2)=f(6)

F is bounded

F is differentiable

F has a root

Continuous functions on closed intervals are always bounded.

Explanation

Continuous functions on closed intervals are always bounded.

10) Let f(x)=4x−7. Which limit property shows its continuity?

Lim f(x)=4a-7

Lim f(x)=0

Lim f(x)=a^2+7

The limit does not exist

Polynomial limit rule applies directly.

Explanation

Polynomial limit rule applies directly.

11) A function is guaranteed continuous on its entire domain if it is

A polynomial

A rational function

A trigonometric function

A logarithmic function

Polynomials are continuous everywhere.

Explanation

Polynomials are continuous everywhere.

12) Let f(x)=5−x. What is the largest interval on which f is continuous?

(-∞,5)

(-∞,5]

[0,5)

[0,5]

Linear functions are continuous everywhere on ℝ.

Explanation

Linear functions are continuous everywhere on ℝ.

13) Suppose f is continuous and f(1)=2, f(4)=5. What must be true?

F(2)=3.5

F is increasing

F(x)=x+1

There exists c ∈ (1,4) such that f(c)=4

IVT guarantees a value 4 between f(1)=2 and f(4)=5.

Explanation

IVT guarantees a value 4 between f(1)=2 and f(4)=5.

14) Let ?. For what value of k is f continuous at x=2?

0

2

4

None

Missing definition prevents determining k.

Explanation

Missing definition prevents determining k.

15) Which statement guarantees that a function f is continuous at a point a?

The function is bounded near a

The lim exists

The function is defined at all points except a

Lim f(a)=f(a)

Continuity requires limit equals function value.

Explanation

Continuity requires limit equals function value.

Fundamentals of Continuity: Limits, One-Sided Limits, and Function Behavior