Problem-Solving Quiz On Continuity, Limits, And Function Analysis

1) A function satisfies lim_{x→1−} f(x)=3, lim_{x→1+} f(x)=3, f(1)=4. What can be concluded?

F is continuous at 1

F has a jump discontinuity at 1

F has a removable discontinuity at 1

No discontinuity occurs

The two one-sided limits agree and give lim_{x→1} f(x)=3, but f(1)=4≠3, so redefining f(1) as 3 would make f continuous: this is a removable discontinuity.

Explanation

The two one-sided limits agree and give lim_{x→1} f(x)=3, but f(1)=4≠3, so redefining f(1) as 3 would make f continuous: this is a removable discontinuity.

2) Given (definition of f involving a parameter k, not specified). For f to be continuous at x=4, what must hold?

K=6

K=0

Both A and C

None of the above

The functional form involving k is not specified, so from the given information no specific value of k can be determined as necessary for continuity at x=4.

Explanation

The functional form involving k is not specified, so from the given information no specific value of k can be determined as necessary for continuity at x=4.

3) Which statements about the continuity of a composite function are correct?

If f is continuous at g(a) and g is continuous at a, then f∘g is continuous at a.

If f∘g is continuous at a, then both f and g are continuous everywhere.

If f is discontinuous, f∘g must be discontinuous.

Continuity of the inside function g matters for continuity of f∘g.

The standard theorem: if g is continuous at a and f is continuous at g(a), then f∘g is continuous at a. The continuity of the inner function g at a is essential; the other statements are false in general.

Explanation

The standard theorem: if g is continuous at a and f is continuous at g(a), then f∘g is continuous at a. The continuity of the inner function g at a is essential; the other statements are false in general.

4) A function f(x) is continuous on [1,5]. Which reasoning is guaranteed?

F(x) takes every value between f(1) and f(5)

F(x) has a minimum but not necessarily a maximum

F(x) may have gaps

Lim_{x→3} f(x) need not exist

By the Intermediate Value Theorem, continuity on a closed interval implies that f takes every value between f(1) and f(5). It also has both a maximum and minimum and cannot have gaps.

Explanation

By the Intermediate Value Theorem, continuity on a closed interval implies that f takes every value between f(1) and f(5). It also has both a maximum and minimum and cannot have gaps.

5) If f is continuous at all irrational numbers but discontinuous at all rational numbers, then f is:

Continuous everywhere

Continuous nowhere

An example of a function with dense discontinuities

A polynomial

Since the rationals are dense in ℝ and f is discontinuous at every rational, its set of discontinuities is dense, while it is still continuous at every irrational.

Explanation

Since the rationals are dense in ℝ and f is discontinuous at every rational, its set of discontinuities is dense, while it is still continuous at every irrational.

6) Suppose f is continuous and f(a)<0<f(b). What conclusion is logically valid?

F is differentiable on [a,b]

F has a root in (a,b)

F(a)=f(b)

F must be monotonic

By the Intermediate Value Theorem, a continuous function that takes opposite signs at a and b must be zero at some c in (a,b).

Explanation

By the Intermediate Value Theorem, a continuous function that takes opposite signs at a and b must be zero at some c in (a,b).

7) Consider a continuous function f. Which reasoning steps prove that f cannot have an infinite jump at a point?

The limit would not exist

The left and right limits must be finite and equal

Continuity forces the limit to equal the function value

Jumps violate the definition of a limit

Continuity at a point requires a finite limit that equals f(a); an infinite jump would destroy the existence of the limit and thus contradict the definition of continuity.

Explanation

Continuity at a point requires a finite limit that equals f(a); an infinite jump would destroy the existence of the limit and thus contradict the definition of continuity.

8) You are told that lim_{x→0} f(x)=2, but the value of f(0) is unknown. What reasoning is correct?

F must be continuous at 0

F(0) must equal 2

The function may or may not be continuous at 0

The limit cannot exist without knowing f(0)

The limit as x→0 depends only on nearby values, not on f(0) itself; continuity at 0 additionally requires f(0)=2, which we do not know.

Explanation

The limit as x→0 depends only on nearby values, not on f(0) itself; continuity at 0 additionally requires f(0)=2, which we do not know.

9) Let f be continuous on [0,4]. If the maximum occurs at a boundary point, which reasoning holds?

The maximum must be at x=0

The maximum must be at an interior point

The maximum could be at x=0 or x=4

The function must be constant

For a continuous function on a closed interval, extrema can occur either at interior critical points or at endpoints; if the maximum is at a boundary, it may be at either endpoint.

Explanation

For a continuous function on a closed interval, extrema can occur either at interior critical points or at endpoints; if the maximum is at a boundary, it may be at either endpoint.

10) A function has a limit at x=a. Which statements must be true for continuity at a?

The function must be defined at a

The limit must equal f(a)

The limit must be finite

The limit must exist

Continuity at a requires that f(a) is defined and that lim_{x→a} f(x) exists as a finite number and equals f(a).

Explanation

Continuity at a requires that f(a) is defined and that lim_{x→a} f(x) exists as a finite number and equals f(a).

11) A continuous function f satisfies: f(1)=1, f(4)=10. Which conclusion is most logically correct using continuity reasoning?

F(2)=5.5

F is linear

There exists a point c∈(1,4) where f(c)=6

F is increasing

By the Intermediate Value Theorem, continuity on [1,4] and values 1 and 10 at the endpoints imply that every value between 1 and 10, including 6, is attained at some c in (1,4).

Explanation

By the Intermediate Value Theorem, continuity on [1,4] and values 1 and 10 at the endpoints imply that every value between 1 and 10, including 6, is attained at some c in (1,4).

12) If f is continuous at x=3, which of the following must be true?

F(3)=0

Lim_{x→3} f(x) = f(3)

F is differentiable at 3

F is linear

Continuity at x=3 means exactly that the limit of f(x) as x→3 exists and equals the function value f(3).

Explanation

Continuity at x=3 means exactly that the limit of f(x) as x→3 exists and equals the function value f(3).

13) For f continuous on (a,b), what conclusions are valid?

F(x) has no jumps

F(x) cannot have removable discontinuities inside (a,b)

The graph of f can be drawn without lifting the pencil

F must be monotonic

Continuity on (a,b) rules out both jump and removable discontinuities, and informally means the graph can be traced without lifting the pencil. Monotonicity is not implied.

Explanation

Continuity on (a,b) rules out both jump and removable discontinuities, and informally means the graph can be traced without lifting the pencil. Monotonicity is not implied.

14) If f(x)=x^2-2x+1, which reasoning justifies its continuity?

All polynomials are continuous

The derivative exists

It has no real roots

The domain is restricted

x^2−2x+1 is a polynomial, and every polynomial is continuous on ℝ, regardless of its roots or derivative.

Explanation

x^2−2x+1 is a polynomial, and every polynomial is continuous on ℝ, regardless of its roots or derivative.

15) Suppose f and g are continuous at a. Which are always continuous at a?

F+g

F−g

Fg (or f/g if g(a)≠0)

|f|

Sums, differences, products, absolute values, and quotients (where the denominator is nonzero at a) of functions continuous at a are themselves continuous at a.

Explanation

Sums, differences, products, absolute values, and quotients (where the denominator is nonzero at a) of functions continuous at a are themselves continuous at a.

Problem-Solving Quiz on Continuity, Limits, and Function Analysis