Understanding the Continuity of a Function

Continuity of a function is a fundamental concept in calculus that describes the behavior of functions as they approach certain points or as their domain is traversed. A function is said to be continuous at a point if there is no interruption in the graph of the function at that point. In other words, you can draw the function at that point without lifting your pencil from the paper.

Definition of Continuity at a Point

A function $f(x)$ is continuous at a point $x = a$ if the following three conditions are met:

1. $f(a)$ is defined (i.e., $a$ is in the domain of $f$).
2. The limit of $f(x)$ as $x$ approaches $a$ exists.
3. The limit of $f(x)$ as $x$ approaches $a$ is equal to $f(a)$.

Mathematically, this can be expressed as:

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

Checking Continuity Over an Interval

A function is continuous on an interval if it is continuous at every point in that interval. This means that the function has no breaks, jumps, or holes in the interval.

Types of Discontinuities

Discontinuities can be classified into different types based on the behavior of the function at the point of discontinuity:

Type of Discontinuity	Description	Graphical Representation
Removable	A hole in the graph; the limit exists, but the function is not defined at that point or $f(a) \neq \lim_{x \to a} f(x)$.	A hollow circle on the graph
Jump	The function has different left and right limits at the point.	A sudden jump in the graph
Infinite	The function approaches infinity as $x$ approaches the point.	A vertical asymptote in the graph
Oscillating	The function oscillates between different values as $x$ approaches the point.	No clear limit as $x$ approaches the point

Formulas and Methods to Check Continuity

To check the continuity of a function at a point $x = a$, we can use the following methods:

1. Direct Substitution: If $f(a)$ is defined and there are no indeterminate forms, simply substitute $x = a$ into the function to find $f(a)$.

2. Limit Evaluation: Evaluate $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$. If both one-sided limits exist and are equal, then $\lim_{x \to a} f(x)$ exists.

3. Piecewise Functions: For piecewise functions, check the continuity of each piece individually and then ensure that the function matches at the boundaries of the pieces.

Examples to Explain Important Points

Example 1: Direct Substitution

Consider the function $f(x) = x^2$. To check if $f(x)$ is continuous at $x = 2$, we can directly substitute $x = 2$ into the function:

$$ f(2) = 2^2 = 4 $$

Since $f(2)$ is defined and there are no indeterminate forms, $f(x)$ is continuous at $x = 2$.

Example 2: Limit Evaluation

Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$. To check if $f(x)$ is continuous at $x = 1$, we evaluate the limit:

$$ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x - 1)(x + 1)}{x - 1} = \lim_{x \to 1} (x + 1) = 2 $$

Since $f(1)$ is not defined (division by zero), but the limit as $x$ approaches 1 exists, the discontinuity at $x = 1$ is removable.

Example 3: Piecewise Functions

Consider the piecewise function:

$$ f(x) = \begin{cases} x^2 & \text{if } x < 2 \ 4 & \text{if } x = 2 \ x + 2 & \text{if } x > 2 \end{cases} $$

To check if $f(x)$ is continuous at $x = 2$, we need to check the limits from the left and right:

$$ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 4 $$

$$ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x + 2) = 4 $$

Since $f(2) = 4$, and both one-sided limits are equal to 4, $f(x)$ is continuous at $x = 2$.

Conclusion

To check the continuity of a function, one must evaluate the function and its limits at the point of interest. Discontinuities can be identified and classified, and understanding the type of discontinuity can help in analyzing the behavior of the function. By using direct substitution, limit evaluation, and careful analysis of piecewise functions, one can determine whether a function is continuous at a given point or over an interval.