Mathematics Quiz on Continuity and Differentiability

What is the definition of continuity at a point c for a function f?

• f is continuous at c if lim (x -> c) f(x) exists

• f is continuous at c if f is differentiable at c

• f is continuous at c if f(c) = 0

• f is continuous at c if lim (x -> c) f(x) = f(c)

Which of the following functions is continuous at x = 0?

• f(x) = 2x + 3

• f(x) = |x|

• f(x) = 1/x, x ≠ 0

• f(x) = 1 if x ≤ 0, 2 if x > 0

What is the left-hand limit of the function f(x) = 1 if x ≤ 0, 2 if x > 0 at x = 0?

• 1

• 2

• 0

• Undefined

Which theorem states that the sum, difference, product, and quotient of continuous functions are continuous?

• Theorem of Limits

• Chain Rule

• Algebra of Continuous Functions

• Fundamental Theorem of Calculus

What is the derivative of the function f(x) = e^x?

• 1

• e^x

• 0

• x * e^x

If f(x) = ln(x), what is f'(x)?

• x

• e^x

• ln(x)

• 1/x

Which of the following is a property of logarithmic functions?

• They have a range of positive real numbers.

• They are ever increasing.

• They are defined for all real numbers.

• They are discontinuous at x = 0.

What is the relationship between differentiability and continuity?

• Every differentiable function is continuous.

• Every continuous function is differentiable.

• Differentiability implies continuity only at endpoints.

• Continuity implies differentiability.

What is the second derivative of f(x) = x^3?

• 6x

• 3x^2

• 3

• 0

Which of the following functions is not continuous at x = 1?

• f(x) = x^2

• f(x) = |x - 1|

• f(x) = 1/x, x ≠ 1

• f(x) = sin(x)