Question

Which of the following set inclusion statements properly describes the relationship between continuous, differentiable, and integrable functions?

a) Continuous ⊆ Differentiable ⊆ Integrable

b) Differentiable ⊆ Continuous ⊆ Integrable

c) Integrable ⊆ Continuous ⊆ Differentiable

d) Integrable ⊆ Differentiable ⊆ Continuous

Answer 1

The correct set inclusion statement that properly describes the relationship between continuous, differentiable, and integrable functions is: Integrable ⊆ Continuous ⊆ Differentiable.

Step-by-step explanation:

The correct set inclusion statement that properly describes the relationship between continuous, differentiable, and integrable functions is: c) Integrable ⊆ Continuous ⊆ Differentiable.

This statement means that all integrable functions are continuous, and all continuous functions are differentiable. However, not all differentiable functions are integrable.

For example, the function f(x) = |x| is integrable and continuous but not differentiable at x = 0. On the other hand, the function g(x) = x^(1/3) is continuous and differentiable but not integrable on the interval [-1, 1] due to an infinite limit at x = 0.