Real Functions
A real function is a mapping that assigns to every element in its domain \(D \subseteq \mathbb{R}\) a unique real number:
\[f : D \to \mathbb{R} \]where \(D\) is a subset of the real numbers. The image of \(f\) is the set of all values \(f(x)\) for \(x \in D\), denoted as \(f(D)\). We can then group these functions to form the set of all real-valued functions with domain \(D\) is denoted as
\[\mathbb{R}^D = \{f : D \to \mathbb{R} \mid D \subseteq \mathbb{R} \} \]For \(f, g \in \mathbb{R}^D\) and \(c \in \mathbb{R}\), we can define the following operations:
- Addition: \((f + g)(x) := f(x) + g(x)\)
- Scalar multiplication: \((c \cdot f)(x) := c \cdot f(x)\)
With these operations, \(\mathbb{R}^D\) forms a vector space over \(\mathbb{R}\). The zero vector in this space is the null function defined by \(f_0(x) = 0\) for all \(x \in D\).
We can also define:
- Product: \((f \cdot g)(x) := f(x) \cdot g(x)\), so the constant function \(f_1(x) = 1\) is the multiplicative identity.
- Quotient: \(\left(\frac{f}{g}\right)(x) := \frac{f(x)}{g(x)}\), defined wherever \(g(x) \neq 0\).
- Composition: For \(f : D \to \mathbb{R}\) and \(g : E \to \mathbb{R}\) with \(f(D) \subseteq E\), \((g \circ f)(x) := g(f(x))\).
Bounded Functions
A function \(f : D \to \mathbb{R}\) is called bounded if its image, i.e the set of its values \(f(D) = {f(x) \mid x \in D}\) is a bounded subset of \(\mathbb{R}\). That is, there exists \(M > 0\) such that
\[|f(x)| \leq M \quad \forall x \in D \]Equivalently, \(f\) is bounded above if there is \(M\) such that \(f(x) \leq M\) for all \(x\), and below if there is \(m\) such that \(f(x) \geq m\) for all \(x\).
The constant function \(f(x) = c\) is bounded for any \(c \in \mathbb{R}\), since \(|f(x)| = |c| \leq M\) for any \(M \geq |c|\).
The identity function \(f(x) = x\) is not bounded on \(\mathbb{R}\), but is bounded if restricted to a bounded domain, e.g., \(D = [a, b]\), where \(a \leq x \leq b\) implies \(|f(x)| \leq \max(|a|, |b|)\).
Monotonic Functions
A function \(f : D \to \mathbb{R}\) is called monotonic if it preserves order in its domain. By preserving order we mean that the function does not change direction in a way that would contradict the order of the inputs. More precisely, for \(x_1, x_2 \in D\) with \(x_1 < x_2\):
- \(f\) is increasing if \(f(x_1) \leq f(x_2)\).
- \(f\) is decreasing if \(f(x_1) \geq f(x_2)\).
- \(f\) is strictly increasing if \(f(x_1) < f(x_2)\).
- \(f\) is strictly decreasing if \(f(x_1) > f(x_2)\).
The constant function \(f(x) = c\) is monotonic (both increasing and decreasing) but not strictly monotonic, since \(f(x_1) = f(x_2) = c\) for all \(x_1, x_2 \in D\).
The identity function \(f(x) = x\) is strictly increasing on \(\mathbb{R}\), but not bounded. If we restrict \(f\) to a bounded interval, say \(f : [a, b] \to \mathbb{R}\), then \(f\) is both strictly increasing and bounded, since for all \(x \in [a, b]\), we have \(a \leq f(x) \leq b\).
Polynomial functions \(f(x) = x^n\) are strictly increasing on \(\mathbb{R}\) if \(n\) is odd (e.g., \(n=1,3,5\)). We can prove this by showing that the derivative \(f'(x) = n x^{n-1}\) is positive for \(x > 0\), negative for \(x < 0\), and zero only at \(x = 0\) if \(n > 1\). Thus, they are strictly increasing on the entire real line.
On the bounded interval \([-1, 1]\), these functions are also bounded, since \(|f(x)| \leq 1\) for all \(x \in [-1, 1]\). However, they are not bounded on \(\mathbb{R}\) as \(x \to \infty\) or \(x \to -\infty\):
\[\forall M > 0, \exists x \in \mathbb{R}: |f(x)| = |x^3| > M \implies x > M^{1/3} \]So for different \(M\), we can find \(x\) such that \(f(x)\) exceeds any bound.