Continuity and Related Concepts in Real Analysis

Continuous Functions

Definition and Intuition

Continuity is a fundamental concept in real analysis and precalculus, describing how small changes in the input of a function produce small changes in the output. Formally, a function f is continuous at a point x_0 if the value of the function at that point matches the limit of the function as x approaches x_0:

• Definition: Let X \subseteq \mathbb{R}, f: X \to \mathbb{R}, and x_0 \in X. f is continuous at x_0 if and only if $\lim_{x \to x_0;\ x \in X} f(x) = f(x_0)$

• Continuous on X: f is continuous on X if it is continuous at every x_0 \in X.

• Discontinuous at x_0: If the above equality fails, f is discontinuous at x_0.

Examples of Continuous and Discontinuous Functions

• Constant Function: f(x) = c is continuous everywhere because $\lim_{x \to x_0} c = c = f(x_0)$.

• Identity Function: f(x) = x is continuous everywhere because $\lim_{x \to x_0} x = x_0 = f(x_0)$.

• Sign Function (sgn): $\operatorname{sgn}(x) = \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \end{cases}$ is continuous everywhere except at 0, where the left and right limits differ.

• Rational/Irrational Indicator: $f(x) = \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q} \end{cases}$ is discontinuous everywhere, as every neighborhood contains both rationals and irrationals.

• Step Function: $f(x) = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}$ is discontinuous at 0, but continuous elsewhere. If restricted to $[0, \infty)$, it is continuous everywhere on that domain.

Equivalent Formulations of Continuity

• Sequential: f is continuous at x_0 if for every sequence a_n \to x_0, $f(a_n) \to f(x_0)$.

• Epsilon-Delta (Open Ball): $\forall \varepsilon > 0, \exists \delta > 0$ such that $|x - x_0| < \delta \implies |f(x) - f(x_0)| < \varepsilon$.

• Epsilon-Delta (Closed Ball): $|x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \varepsilon$.

Properties of Continuous Functions

• Arithmetic Operations: If f and g are continuous at x_0, so are f+g, f-g, fg, max(f,g), min(f,g), and f/g (if g(x) \neq 0).

• Composition: If f is continuous at x_0 and g is continuous at f(x_0), then g \circ f is continuous at x_0.

• Examples: Polynomials, exponentials, and absolute value functions are continuous on their natural domains.

Left and Right Limits

Definitions and Motivation

One-sided limits help analyze function behavior when approaching a point from only one direction. This is crucial for understanding discontinuities and piecewise functions.

• Right Limit: $f(x_0+) = \lim_{x \to x_0;\ x > x_0} f(x)$

• Left Limit: $f(x_0-) = \lim_{x \to x_0;\ x < x_0} f(x)$

• These may exist even if f(x_0) is undefined.

Continuity and One-Sided Limits

• If f is continuous at x_0, then $f(x_0+) = f(x_0) = f(x_0-)$.

• If both one-sided limits exist and equal f(x_0), then f is continuous at x_0.

Types of Discontinuities

• Jump Discontinuity: One-sided limits exist but are not equal.

• Removable Discontinuity: One-sided limits exist and are equal, but not equal to f(x_0) (or f(x_0) is undefined).

• Asymptotic Discontinuity: Function approaches infinity from one or both sides (e.g., $f(x) = 1/x$ at 0).

• Oscillatory Discontinuity: No limit exists due to wild oscillation (e.g., rational/irrational indicator).

The Maximum Principle

Boundedness and Extrema

Continuous functions on closed intervals have special properties due to compactness:

• Boundedness: If f is continuous on $[a, b]$, then $\exists M$ such that $|f(x)| \leq M$ for all $x \in [a, b]$.

• Global Maximum/Minimum: f attains its maximum and minimum on $[a, b]$:

  • $\exists x_{max} \in [a, b]$ such that $f(x_{max}) \geq f(x)$ for all $x \in [a, b]$

  • $\exists x_{min} \in [a, b]$ such that $f(x_{min}) \leq f(x)$ for all $x \in [a, b]$

Note: The maximum or minimum may not be unique.

The Intermediate Value Theorem (IVT)

Statement and Consequences

The IVT is a cornerstone of real analysis, ensuring that continuous functions on closed intervals take on every value between their minimum and maximum.

• Theorem: If f is continuous on $[a, b]$ and $y$ is between $f(a)$ and $f(b)$, then $\exists c \in [a, b]$ such that $f(c) = y$.

• Applications: Existence of roots (e.g., $\sqrt{2}$), and showing that continuous functions cannot "jump" over values.

• IVT guarantees existence, not uniqueness: A value may be taken at multiple points.

Monotonic Functions

Definitions and Properties

• Monotone Increasing: $y > x \implies f(y) \geq f(x)$

• Strictly Monotone Increasing: $y > x \implies f(y) > f(x)$

• Monotone Decreasing: $y > x \implies f(y) \leq f(x)$

• Strictly Monotone Decreasing: $y > x \implies f(y) < f(x)$

• Constant functions are both monotone increasing and decreasing.

Relationship to Continuity

• Continuity does not imply monotonicity (e.g., $f(x) = x^2$).

• Monotonicity does not imply continuity (e.g., step functions).

• Strictly monotone and continuous functions on $[a, b]$ are invertible, and their inverses are also continuous and strictly monotone.

Uniform Continuity

Definition and Motivation

Uniform continuity strengthens the concept of continuity by requiring that the same $\delta$ works for all points in the domain, not just locally.

• Definition: f is uniformly continuous on X if $\forall \varepsilon > 0, \exists \delta > 0$ such that $|x - x_0| < \delta \implies |f(x) - f(x_0)| < \varepsilon$ for all $x, x_0 \in X$.

• Every uniformly continuous function is continuous, but not every continuous function is uniformly continuous.

• Example: $f(x) = 1/x$ is continuous on $(0, 2)$ but not uniformly continuous.

• On closed intervals: Every continuous function on $[a, b]$ is uniformly continuous (Heine–Borel property).

Limits at Infinity

Definitions and Examples

• Adherent Point at Infinity: $+\infty$ is adherent to $X$ if $\forall M \in \mathbb{R}, \exists x \in X$ with $x > M$ (i.e., $X$ is unbounded above).

• Limit at Infinity: $\lim_{x \to +\infty;\ x \in X} f(x) = L$ if $\forall \varepsilon > 0, \exists M$ such that $x > M \implies |f(x) - L| < \varepsilon$.

• Example: $\lim_{x \to +\infty} 1/x = 0$ on $(0, \infty)$.