Limits and Continuity - Key Properties

Continuity is one of the most fundamental concepts in analysis, formalizing the intuitive idea that a function has no breaks, jumps, or holes in its graph. This concept is central to understanding the behavior of functions and enables powerful theorems about their properties.

DefinitionContinuity at a Point

A function $f$ is continuous at a point $a$ if the following three conditions hold:

$f(a)$ is defined
$\lim_{x \to a} f(x)$ exists
$\lim_{x \to a} f(x) = f(a)$

If any of these conditions fails, we say $f$ is discontinuous at $a$ .

The definition encapsulates the idea that we can evaluate the limit by simply substituting: the limiting behavior matches the actual value. This seemingly simple requirement has profound consequences throughout mathematics.

DefinitionTypes of Discontinuities

If $f$ is discontinuous at $a$ , we classify the discontinuity as:

Removable discontinuity: $\lim_{x \to a} f(x)$ exists but either $f(a)$ is undefined or $\lim_{x \to a} f(x) \neq f(a)$
Jump discontinuity: Both one-sided limits exist but $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$
Infinite discontinuity: At least one one-sided limit is infinite
Essential discontinuity: The limit does not exist in any sense (neither finite nor infinite)

ExampleAnalyzing Discontinuities

Consider the function $f(x) = \begin{cases} \frac{x^2 - 4}{x - 2} & x \neq 2 \\ 3 & x = 2 \end{cases}$

For $x \neq 2$ , we can simplify: $f(x) = \frac{(x-2)(x+2)}{x-2} = x + 2$ . Thus, $\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 4$

Since $f(2) = 3 \neq 4$ , the function has a removable discontinuity at $x = 2$ . We could "remove" it by redefining $f(2) = 4$ .

DefinitionContinuity on an Interval

A function $f$ is continuous on an open interval $(a, b)$ if it is continuous at every point in the interval.

For a closed interval $[a, b]$ , we say $f$ is continuous on $[a, b]$ if:

$f$ is continuous on $(a, b)$
$\lim_{x \to a^+} f(x) = f(a)$ (right-continuous at $a$ )
$\lim_{x \to b^-} f(x) = f(b)$ (left-continuous at $b$ )

Remark

Elementary functions (polynomials, rational functions, trigonometric functions, exponential and logarithmic functions) are continuous wherever they are defined. Compositions and arithmetic combinations of continuous functions are also continuous, which allows us to establish continuity for complex expressions without returning to the epsilon-delta definition each time.

The concept of continuity bridges local properties (behavior at a point) with global properties (behavior on intervals), enabling theorems like the Intermediate Value Theorem and Extreme Value Theorem.