Continuity (Real Functions)#
Definition: Continuity of Real Functions
A real function \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) is continuous at \(c \in \mathbb{R}\) if and only if \(f\)'s [limit](./Limits%20(Real%20Functions.md#Real%20Limits) at \(c\) is equal to \(f(c)\):
If \(f\) is continuous at every \(c \in S \subseteq \mathcal{D}\), then we say that \(f\) is continuous on \(S\). If \(S = \mathcal{D}\), then we simply say that \(f\) is continuous.
Example: \(\frac{(x-1)(x+3)}{x - 2}\)
The function \(f: \mathbb{R} \setminus \{2\} \to \mathbb{R}\) defined as
is continuous. We do not require that \(\lim_{x \to 2} f(x) = f(2)\), since \(f\) is not defined for \(x = 2\).
Example: Polynomial Functions
All real polynomial functions are continuous on all of \(\mathbb{R}\).
Example: Exponential Function
The real exponential function is continuous on all of \(\mathbb{R}\)..
Example: Trigonometric Functions
All real trigonometric functions are continuous (of course, only where they are defined).
Theorem: Operations with Continuous Functions
Let \(f: \mathcal{D}_f \subseteq \mathbb{R} \to \mathbb{R}\) and \(g: \mathcal{D}_g \subseteq \mathbb{R} \to \mathbb{R}\) be real functions and let \(c \in \mathbb{R}\).
If \(f\) and \(g\) are continuous at \(c\), then so are:
- \(\alpha f + \beta g\) for all \(\alpha,\beta \in \mathbb{R}\);
- \(f \cdot g\);
- \(f / g\) provided that \(g(c) \ne 0\);
- \(f \circ g\) provided that \(g(\mathcal{D}_g) \subseteq \mathcal{D}_f\).
Proof
TODO
The Extreme Value Theorem
Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \([a;b] \subseteq \mathcal{D}\) be a closed interval.
If \(f\) is continuous on \([a;b]\), then there exist at least one \(x_{\text{of min}} \in [a;b]\) and at least one \(x_{\text{of max}} \in [a;b]\) such that
Intuition
This theorem says that if a function is continuous on a closed interval, then it has a minimum and a maximum value on it.
TODO: Add diagram
Proof
TODO
The Intermediate Value Theorem
Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \([a;b] \subseteq \mathcal{D}\) be a closed interval.
If \(f\) is continuous on \([a;b]\), then for each \(y \in (\min\{f(a), f(b)\}; \max\{f(a), f(b)\})\), there exists at least one \(x \in [a;b]\) such that \(f(x) = y\).
Intuition
The theorem says that if a function is continuous on a closed interval, then it must generate all values between its minimum and maximum value on said interval.
Proof
TODO
Bolzano's Theorem
Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \([a;b] \subseteq \mathcal{D}\) be a closed interval.
If \(f\) is continuous on \([a;b]\) and \(f(a) \cdot f(b) \lt 0\), then there exists at least one \(x \in (a;b)\) such that \(f(x) = 0\).
Proof
This is just a special case of the intermediate value theorem.
Theorem: Fixed-Point Theorem
Let \([a;b] \subset \mathbb{R}\) be a closed interval and let \(f: [a; b] \to [a;b]\) be a real function.
If \(f\) is continuous, then there exists at least one \(\xi \in [a;b]\) with \(f(\xi) = \xi\).
Proof
TODO
Theorem: Riemann Integrability of Continuous Functions
Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function, let \([a, b] \subseteq \mathcal{D}\) be a closed interval and let \(X \subseteq [a,b]\) be the set of all \(x \in [a;b]\) at which \(f\) is discontinuous.
If \(f\) is bounded on \([a,b]\) and \(X\) has Lebesgue measure zero, then \(f\) is Riemann-integrable on \([a,b]\).
Tip: Continuity \(\implies\) Riemann-Integrability
A direct consequence of this is that if \(f\) is continuous on \([a,b]\), then it is also Riemann-integrable on \([a,b]\).
Proof
TODO
Theorem: Antidifferentiability of Continuous Function
Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function.
If \(f\) is continuous on \(S \subseteq \mathcal{D}\), then it is also antidifferentiable on \(S\).
Proof
TODO
Continuous Extension#
Definition: Continuous Extension
Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and \(p \in \mathbb{R}\) be a limit point of \(\mathcal{D}\).
A continuous extension of \(f\) at \(p\) is any real function \(\tilde{f}: \mathcal{D} \cup \{p\} \to \mathbb{R}\) which is continuous at \(p\) with \(\tilde{f}(x) = f(x)\) for all \(x \in \mathcal{D} \setminus \{p\}\).