Contents (Click to skip to that section):
- What is a Continuous Function?
- Different Types of Continuity
- List of Common Continuous Functions.
- How To Check for The Continuity of a Function.
In layman’s terms, a continuous function is one that you can draw without taking your pencil from the paper. If you have holes, jumps, or vertical asymptotes, you will have to lift your pencil up and so do not have a continuous function.
More formally, a function (f) is continuous if, for every point x = a:
- The function is defined at a. In other words, point a is in the domain of f,
- The limit of the function exists at that point, and is equal as x approaches a from both sides,
- The limit of the function, as x approaches a, is the same as the function output (i.e. the y-value) at a.
If a function is simply “continuous” without any further information given, then you would generally assume that the function is continuous everywhere (i.e. the set of all real numbers from -∞ to + ∞). However, sometimes a particular piece of a function can be continuous, while the rest may not be.
- Continuous on an interval: A function f is continuous on an interval if it is continuous at every point in the interval. For example, you could define your interval to be from -1 to +1. As long as the function is continuous in that little area, then you can say it’s continuous on that specific interval.
- Continuity at an endpoint: There are two possible endpoints for a function: all the way to the left (in the far negative direction), and all the way to the right (in the far positive direction). Assuming the limit exists, “continuity at an endpoint” means that the function is continuous from the right (for the left endpoint) or continuous from the left (for the right endpoint).
A left-continuous function is continuous for all points from only one direction (when approached from the left).
It is a function defined up to a certain point, c, where:
- The function is defined on an closed interval [d, c], lying to the left of c,
- The limit at that point, c, equals the function’s value at that point.
The following image shows a left continuous function up to the point x = 4:
Note how the function value, at x = 4, is equal to the function’s limit as the function approaches the point from the left.
Formal Definition of a Left-Continuous Function
Formally, a left-continuous function f is left-continuous at point c if
limx→c–f(x) = f(c).
In other words, f(x) approaches c from below, or from the left, or for x < c (Morris, 1992). The right-continuous function is defined in the same way (replacing the left hand limit c- with the right hand limit c+ in the subscript).
A right continuous function is defined up to a certain point. The following image shows a right continuous function up to point, x = 4:
Note that the point in the above image is filled in. On a graph, this tells you that the point is included in the domain of the function. If the point was represented by a hollow circle, then the point is not included in the domain (just every point to the right of it, in this graph) and the function would not be right continuous.
Note that this type of continuity is defined for a point, not for an entire function.
More Formal Definition of a Right Continuous Function
The reason why the function isn’t considered right continuous is because of how these functions are formally defined. Two conditions must be true about the behavior of the function as it leads up to the point:
- exists: The limit of functions must exist at the point. The + sign above the “a” means that the point is being approached from the positive end of the number line; In other words, it’s approaching from the right.
- The right hand limit leading up to the point a, must equal the limit of the function at that point.
In the second example above, the circle was hollowed out, indicating that the point isn’t included in the domain of the function. As the point doesn’t exist, the limit at that point doesn’t exist either.
These Aren’t Really Continuous Functions!
The label “right continuous function” is a little bit of a misnomer, because these are not continuous functions. In order for a function to be continuous, the right hand limit must equal f(a) and the left hand limit must also equal f(a). The definition for a right continuous function mentions nothing about what’s happening on the left side of the point. The function may be continuous there, or it may not be. The only way to know for sure is to also consider the definition of a left continuous function.
A uniformly continuous function on a given set A is continuous at every point on A. The way this is checked is by checking the neighborhoods around every point, defining a small region where the function has to stay inside.
More formally, this is stated as:
A function f : A → ℝ is uniformly continuous on A if, for every number ε > 0, there is a δ > 0; whenever x, y ∈ A and |x − y| < δ it follows that |f(x) − f(y)| < ε.
What that formal definition is basically saying is choose some values for ε, then find a δ that works for all of the x-values in the set. If the same values work, the function meets the definition.
Graph of a Uniformly Continuous Function
Graphically, look for points where a function suddenly increases or decreases curvature. The definition doesn’t allow for these large changes; It’s very unlikely you’ll be able to create a “box” of uniform size that will contain the graph. The function might be continuous, but it isn’t uniformly continuous.
The game here is to try and find a uniform box of height x width = 2ε x 2 δ that, when moved, will keep the graph contained within the edges of the box. So, in a way, the “uniform” part of the definition refers to a “box of uniform size”.
Properties of a Uniformly Continuous Function
These functions share some common properties.
- All of these functions are bounded on a closed interval [a, b] and will achieve a maximum in the set (a, b).
- Every uniformly continuous function is also a continuous function. However, not all continuous functions are uniformly continuous. Therefore, you can think of a these function as ones that are “more” continuous.
- They may or may not be differentiable. Uniform continuity doesn’t necessarily imply differentiability.
All of the following functions are continuous:
- Absolute value function,
- Exponential function,
- Logarithmic function,
- All polynomial functions,
- All Power Functions,
- All rational functions,
- All trigonometric functions.
- The natural exponential function ex (where e is Euler’s number)
There are a few general rules you can refer to when trying to determine if your function is continuous. For other functions, you need to do a little detective work.
Step 1: Draw the graph with a pencil to check for the continuity of a function. If your pencil stays on the paper from the left to right of the entire graph, without lifting the pencil, your function is continuous. In other words, if your graph has gaps, holes or is a split graph, your graph isn’t continuous. If you aren’t sure about what a graph looks like if it’s not continuous, check out the images in this article: When is a Function Not Differentiable?
Step 2: Figure out if your function is listed in the List of Continuous Functions. If it is, then there’s no need to go further; your function is continuous.
Step 3: Check if your function is the sum (addition), difference (subtraction), or product (multiplication) of one of the continuous functions listed in Step 2. If it is, your function is continuous. For example, sin(x) * cos(x) is the product of two continuous functions and so is continuous.
Step 4: Check your function for the possibility of zero as a denominator. The ratio f(x)/g(x) is continuous at all points x where the denominator isn’t zero. In other words, there’s going to be a gap at x = 0, which means your function is not continuous.
Image: By Eskil Simon Kanne Wadsholt – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=50614728
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