Properties of Continuity and Intermediate Value Theorem

A key property of continuous probability distributions is that **the probability of a random variable taking on any single, exact value is zero.** That is, for any specific value 'a', P(X = a) = 0.

In continuous distributions, probability is defined over intervals, represented by the area under the probability density function (PDF) curve. Since a single point on the x-axis has no width, the "area" above a single point is zero, hence the probability of the variable landing on any *exact* value is zero.

Because the probability of a single value is zero, including or excluding the endpoint 'a' does not change the probability for a continuous variable. Therefore:

• P(X ≤ a) = P(X < a)
• P(X ≥ a) = P(X > a)
• P(a ≤ X ≤ b) = P(a < X < b)

The Intermediate Value Theorem is a fundamental theorem in calculus about the values that a continuous function can take. It states that if a function f is continuous on a closed interval [a, b], and N is any number between f(a) and f(b) (where f(a) ≠ f(b)), then there must exist at least one number c in the open interval (a, b) such that f(c) = N.

In simpler terms, if you can draw the graph of a continuous function between two points without lifting your pen, then the function must pass through every y-value between the y-values of those two points.

The Intermediate Value Theorem has two essential conditions that must be met:

1. The function f must be **continuous** on the closed interval [a, b].
2. The value N must be **between** f(a) and f(b) (not equal to either, unless f(a)=f(b)).

If either of these conditions is not met, the theorem does not guarantee the existence of the value c.

A common application of the IVT is to show that a function has a root (or zero) within a specific interval. To do this:

1. Verify the function is continuous on the closed interval [a, b].
2. Evaluate the function at the endpoints, f(a) and f(b).
3. If f(a) and f(b) have opposite signs (one is positive and one is negative), then 0 is a value between f(a) and f(b).

By the IVT, since 0 is between f(a) and f(b), there must exist a value c between a and b such that f(c) = 0. This proves that a root exists within that interval.

The IVT's guarantee fails if the conditions are not met:

• **Discontinuous Function:** If the function is not continuous on the closed interval [a, b], it might "jump" over the intermediate value N without ever reaching it.
• **Value N is not between f(a) and f(b):** If N is greater than both f(a) and f(b) (or less than both), the theorem doesn't say anything about whether the function hits that value between 'a' and 'b'.

Also, the IVT **only guarantees existence**; it does not provide a method to find the exact value of 'c'.