Unit V: Continuous and Differentiable Functions on Intervals

Continuous Functions on Intervals

Continuous functions defined on closed and bounded intervals possess unique properties that do not necessarily hold on other types of intervals. These properties guarantee the existence of bounds, extreme values, and intermediate states.

Boundedness Theorem

The Boundedness Theorem states that a continuous function on a closed and bounded interval cannot grow infinitely large in either the positive or negative direction.

Boundedness Theorem: Let I = [a, b] be a closed and bounded interval, and let f: I → ℝ be a continuous function on I. Then f is bounded on I. That is, there exists a real number M > 0 such that |f(x)| ≤ M for all x ∈ I.

Step-by-Step Proof Structure (by Contradiction):

1. Assume that f is not bounded on [a, b].
2. Then, for each natural number n ∈ ℕ, there exists an element x_n ∈ [a, b] such that |f(x_n)| > n.
3. This constructs a sequence (x_n) in the bounded interval [a, b]. By the Bolzano-Weierstrass Theorem, there exists a subsequence (x_nk) that converges to some real number c ∈ [a, b].
4. Since f is continuous on [a, b], it must be continuous at c. Therefore, the sequence of values f(x_nk) must converge to f(c).
5. A convergent sequence of real numbers must be bounded. However, this contradicts our construction where |f(x_nk)| > n_k ≥ k for all k.
6. Thus, the initial assumption must be false, meaning f is bounded on [a, b].

Common Mistake: Forgetting the requirement that the interval must be closed and bounded. For example, the function f(x) = 1/x is continuous on the open interval (0, 1) but is not bounded.

Maximum-Minimum Theorem

This theorem (often called the Extreme Value Theorem) guarantees that a continuous function on a compact interval reaches its absolute limits.

Maximum-Minimum Theorem: Let I = [a, b] be a closed and bounded interval, and let f: I → ℝ be continuous on I. Then f attains its absolute maximum and absolute minimum values on I. That is, there exist points x_* and x^* in I such that:
f(x_*) ≤ f(x) ≤ f(x^*) for all x ∈ I.

Important Observation: The points x_* and x^* do not have to be unique. For instance, the constant function f(x) = 5 on [1, 3] attains its maximum and minimum at every single point in the interval.

Location of Roots Theorem

This theorem (also known as Bolzano's Theorem) provides a reliable method for verifying the existence of a zero (or root) of a continuous function.

Location of Roots Theorem: Let I = [a, b] be a closed and bounded interval, and let f: I → ℝ be continuous on I. If f(a) and f(b) have opposite signs (i.e., f(a) < 0 < f(b) or f(b) < 0 < f(a)), then there exists at least one point c ∈ (a, b) such that f(c) = 0.

Practical Example: Prove that the equation x³ - 3x + 1 = 0 has a root in the interval [0, 1].
Let f(x) = x³ - 3x + 1. Since f is a polynomial, it is continuous everywhere, including on [0, 1].
Evaluating the endpoints:
f(0) = 0³ - 3(0) + 1 = 1 > 0
f(1) = 1³ - 3(1) + 1 = -1 < 0
Since f(0) > 0 and f(1) < 0, by the Location of Roots Theorem, there exists some c ∈ (0, 1) where f(c) = 0.

Intermediate Value Theorem (IVT)

The Intermediate Value Theorem is a generalization of the Location of Roots Theorem, stating that continuous functions cannot skip values.

Intermediate Value Theorem: Let I = [a, b] be an interval, and let f: I → ℝ be continuous on I. If k is any real number between f(a) and f(b), then there exists at least one point c ∈ (a, b) such that f(c) = k.

Real-World Application: If you climb a mountain from an altitude of 1,000 meters to 3,000 meters, you must cross exactly 2,000 meters of altitude at some point during your journey, assuming your path is continuous.

Preservation of Intervals Theorem

Continuous functions preserve the structural connectedness of intervals.

Preservation of Intervals Theorem: If I is an interval and f: I → ℝ is continuous on I, then the image set f(I) is also an interval.

Important Note: The theorem does not guarantee that the type of interval is preserved. For instance, the continuous function f(x) = x² maps the open interval (-1, 1) to the half-open interval [0, 1).

Differentiability of Functions

Differentiability represents the rate of change and local linearity of a function.

Definition: Let I be an interval containing the point c. A function f: I → ℝ is said to be differentiable at c if the limit:
f'(c) = lim_x→c (f(x) - f(c)) / (x - c)
exists. The value f'(c) is called the derivative of f at c.

Caratheodory's Theorem

Caratheodory's Theorem provides an alternative formulation of differentiability that avoids fractions and division-by-zero difficulties, making it highly effective for chain rule proofs.

Caratheodory's Theorem: Let f be defined on an interval I containing the point c. Then f is differentiable at c if and only if there exists a function φ defined on I that is continuous at c and satisfies:
f(x) - f(c) = φ(x)(x - c) for all x ∈ I.
In this case, φ(c) = f'(c).

Interior-Extremum Theorem

Also known as Fermat's Theorem, this result establishes the necessity of a zero derivative at local extrema occurring inside an open interval.

Interior-Extremum Theorem: Let c be an interior point of the interval I where f: I → ℝ has a relative (local) extremum. If f is differentiable at c, then f'(c) = 0.

Common Mistake: Believing that f'(c) = 0 automatically means c is a relative extremum. Consider f(x) = x³ at c = 0. Here f'(0) = 0, but the function has neither a maximum nor a minimum at 0; it is an inflection point.

Rolle's Theorem

Rolle's Theorem states that a continuous, differentiable function that starts and ends at the same value must have a flat point (a horizontal tangent line) somewhere in between.

Rolle's Theorem: Let f: [a, b] → ℝ be continuous on [a, b] and differentiable on (a, b). If f(a) = f(b), then there exists at least one point c ∈ (a, b) such that f'(c) = 0.

Applying Rolle's Theorem:
Let f(x) = x² - 4x + 3 on [1, 3].
1. f is a polynomial, so it is continuous on [1, 3] and differentiable on (1, 3).
2. f(1) = 1 - 4 + 3 = 0, and f(3) = 9 - 12 + 3 = 0. Since f(1) = f(3),
Rolle's Theorem guarantees some c ∈ (1, 3) where f'(c) = 2c - 4 = 0. Solving this yields c = 2, which lies in (1, 3).

Mean Value Theorem

The Mean Value Theorem (MVT) generalizes Rolle's Theorem by tilting the boundary values.

Mean Value Theorem: Let f: [a, b] → ℝ be continuous on [a, b] and differentiable on (a, b). Then there exists at least one point c ∈ (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a).

Geometric Interpretation: There is some point c where the instantaneous slope (tangent line) is parallel to the average slope (secant line) connecting the interval's endpoints.

Darboux's Theorem

While derivative functions (f') do not have to be continuous themselves, they still possess the Intermediate Value Property. This is known as Darboux's Theorem.

Darboux's Theorem: If f is differentiable on [a, b] and if k is any number between f'(a) and f'(b), then there exists at least one point c ∈ (a, b) such that f'(c) = k.

Important Observation: Because of Darboux's Theorem, a derivative function f' can never have a simple "jump" discontinuity. If f' has a discontinuity, it must be of an essential (oscillatory) nature, such as the derivative of f(x) = x² sin(1/x) at x = 0.