Basic Concepts of Continuity

Continuity means the graph of a function has no breaks. Grasping the concept is essential for learning calculus.

Definition of Continuity

Definition of continuity at a point

Mathematical description

$f(x)$ is continuous at $x_0$ if and only if:

1. $f(x_0)$ exists (the function is defined at $x_0$)
2. $\lim_{x \to x_0} f(x)$ exists
3. $\lim_{x \to x_0} f(x) = f(x_0)$

Epsilon–delta definition

$f(x)$ is continuous at $x_0$ iff for any $\varepsilon > 0$ there exists $\delta > 0$ such that when $|x - x_0| < \delta$, we have $|f(x) - f(x_0)| < \varepsilon$.

Geometric Meaning of Continuity

Intuitive view

Geometrically, continuity means the curve has no jumps or breaks at that point.

• Continuous point: the curve stays connected at the point
• Discontinuous point: the curve has a jump or gap

Graph features

1. Continuous functions: the graph is a smooth connected curve
2. Discontinuous functions: the graph has jumps or gaps at some points

Continuous function example

Discontinuous function example

Basic Properties of Continuity

1. Local Properties

• Local boundedness: if $f$ is continuous at $x_0$, it is bounded in a neighborhood of $x_0$
• Local sign preservation: if $f$ is continuous at $x_0$ and $f(x_0) > 0$, then $f$ stays positive in some neighborhood of $x_0$

2. Operational Properties

If $f(x)$ and $g(x)$ are continuous at $x_0$, then:

1. Sum/difference: $f(x) \pm g(x)$ is continuous at $x_0$
2. Product: $f(x) \cdot g(x)$ is continuous at $x_0$
3. Quotient: $\frac{f(x)}{g(x)}$ is continuous at $x_0$ if $g(x_0) \neq 0$
4. Composition: $f(g(x))$ is continuous at $x_0$ if $g$ is continuous at $x_0$ and $f$ is continuous at $g(x_0)$

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Exercises

Exercise 1

Determine whether $f(x) = x^2 + 1$ is continuous at $x = 2$.

Exercise 2

Check the continuity of $f(x) = \dfrac{1}{x}$ at $x = 0$.

Exercise 3

For $f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases}$, determine continuity at $x = 1$.

Exercise 4

For $f(x) = \begin{cases} x^2, & x \leq 0 \\ 2x + 1, & x > 0 \end{cases}$, decide continuity at $x = 0$.

Exercise 5

Given $f(x)$ is continuous at $x = a$ and $f(a) > 0$, prove that there exists $\delta > 0$ such that when $|x - a| < \delta$, $f(x) > 0$.

Exercise 6

Discuss continuity of $f(x) = \dfrac{\sin x}{x}$ at $x = 0$.

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Summary

Symbols Used in This Article