Limits and Continuity - Applications

The Intermediate Value Theorem is one of the most important consequences of continuity, establishing that continuous functions on closed intervals take on all intermediate values. This theorem has profound theoretical and practical applications.

TheoremIntermediate Value Theorem (IVT)

Let $f$ be continuous on the closed interval $[a, b]$ and let $N$ be any number between $f(a)$ and $f(b)$ , where $f(a) \neq f(b)$ . Then there exists at least one number $c$ in $(a, b)$ such that $f(c) = N$

Geometrically, this says that the graph of a continuous function cannot jump over any value without passing through it. The theorem guarantees existence but not uniqueness—there may be multiple points $c$ where $f(c) = N$ .

ExampleProving Existence of Roots

Show that the equation $x^5 + x^3 + 1 = 0$ has at least one real solution.

Let $f(x) = x^5 + x^3 + 1$ . This polynomial is continuous everywhere. Evaluate:

$f(-2) = (-2)^5 + (-2)^3 + 1 = -32 - 8 + 1 = -39 < 0$
$f(0) = 1 > 0$

Since $f(-2) < 0 < f(0)$ and $f$ is continuous on $[-2, 0]$ , the IVT guarantees there exists $c \in (-2, 0)$ with $f(c) = 0$ . Thus the equation has at least one real root.

Remark

The IVT requires all three conditions:

$f$ must be continuous on $[a, b]$
The interval must be closed
$N$ must lie strictly between $f(a)$ and $f(b)$

If any condition fails, the theorem may not hold.

TheoremExtreme Value Theorem (EVT)

If $f$ is continuous on a closed interval $[a, b]$ , then $f$ attains both an absolute maximum and an absolute minimum on $[a, b]$ . That is, there exist points $c, d \in [a, b]$ such that $f(c) \leq f(x) \leq f(d) \quad \text{for all } x \in [a, b]$

The EVT asserts that a continuous function on a closed, bounded interval is bounded and attains its bounds. This is not true for open intervals or discontinuous functions.

ExampleExtreme Value Theorem Application

Consider $f(x) = x^2$ on $[-1, 2]$ . Since $f$ is continuous (polynomials are continuous everywhere), the EVT guarantees maximum and minimum values exist.

Computing: $f(-1) = 1$ , $f(0) = 0$ , $f(2) = 4$ .

The minimum value is $f(0) = 0$ and the maximum value is $f(2) = 4$ .

However, on the open interval $(-1, 2)$ , while $f$ is still continuous, it has no maximum value on this interval—the values approach but never reach 4.

Remark

The IVT and EVT are existence theorems—they guarantee that certain values exist without providing methods to find them explicitly. These theorems are fundamental to optimization theory, numerical analysis, and the theoretical foundations of calculus.

Together, these theorems demonstrate the power of continuity in establishing global properties of functions from local information.