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

Question 1

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

Accepted Answer

All polynomials are continuous

Answer

The derivative exists

Answer

It has no real roots

Answer

The domain is restricted

Question 2

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

Accepted Answer

f(0) must equal 2

Answer

f must be continuous at 0

Answer

The function may or may not be continuous at 0

Answer

The limit cannot exist without knowing f(0)

Question 3

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

Accepted Answer

f+g

Answer

f−g

Answer

fg (or f/g if g(a)≠0)

Answer

|f|

Question 4

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

Accepted Answer

f is linear

Answer

f(2)=5.5

Answer

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

Answer

f is increasing

Question 5

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

Accepted Answer

lim_{x→3} f(x) = f(3)

Answer

f(3)=0

Answer

f is differentiable at 3

Answer

f is linear

Question 6

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

Accepted Answer

f(x) has no jumps

Answer

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

Answer

The graph of f can be drawn without lifting the pencil

Answer

f must be monotonic

Question 7

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 8

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

Accepted Answer

None of the above

Answer

k=6

Answer

k=0

Answer

Both A and C

Question 9

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

Accepted Answer

The function must be defined at a

Answer

The limit must equal f(a)

Answer

The limit must be finite

Answer

The limit must exist

Question 10

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

Accepted Answer

Continuity forces the limit to equal the function value

Answer

The limit would not exist

Answer

The left and right limits must be finite and equal

Answer

Jumps violate the definition of a limit

Question 11

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

Accepted Answer

Continuous everywhere

Answer

Continuous nowhere

Answer

An example of a function with dense discontinuities

Answer

A polynomial

Question 12

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

Accepted Answer

f has a root in (a,b)

Answer

f is differentiable on [a,b]

Answer

f(a)=f(b)

Answer

f must be monotonic

Question 13

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

Accepted Answer

f has a removable discontinuity at 1

Answer

f is continuous at 1

Answer

f has a jump discontinuity at 1

Answer

No discontinuity occurs

Question 14

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

Accepted Answer

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

Answer

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

Answer

f(x) may have gaps

Answer

lim_{x→3} f(x) need not exist

Question 15

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

Accepted Answer

The function must be constant

Answer

The maximum must be at x=0

Answer

The maximum must be at an interior point

Answer

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

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