Continuous

A function is said to be continuous if its graph has no sudden breaks or jumps. If a function is continuous, we can trace its graph without ever lifting our pencil. Below are some examples of continuous functions:

Examples

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Sometimes, a function is only continuous on certain intervals. For example, the function,

is only continuous on the intervals (-∞, -1), (-1, 1), and (1, ∞). This is because at x = ±1, f has vertical asymptotes, which are breaks in the graph (you can also think think of vertical asymptotes as infinite jumps). The function is not defined when x = 1 or -1. Refer to the graph below:

Example

Note: Another way of saying that a function is continuous everywhere is to say that it is continuous on the interval (-∞, ∞).

Below is a function, f, that is discontinuous at x = 2 because the graph suddenly jumps from 2 to 3. The closed dot at (2, 3) means that the function value is actually 3 at x = 2. This can be written as f(2) = 3. The open dot at (2, 2) means that the function value approaches 2 as you draw the graph from the left, but the function value is not actually 2 at x = 2 (f(2) ≠ 2).

Below is another example of a discontinuous function. The function is discontinuous at x = 1 because it has a hole in it. Though we may think that the function value should be ½ at x = 1 the value is actually 1. This can be written as f(1) = 1 ≠ ½. The function approaches ½ as x gets close to 1 from the right and the left, but suddenly jumps to 1 when x is exactly 1:

Important but subtle point on discontinuities: A function that is not continuous at a certain point is not necessarily discontinuous at that point. For example, the function,

is not continuous at x = -1 or 1 because it has vertical asymptotes at those points. However, it is not technically correct to say that is discontinuous at x = -1 or 1, because is not even defined at those x values! We say that is continuous everywhere on its domain. On the other hand, the functions with jumps in the last 2 examples are truly discontinuous because they are defined at the jump.


formal definition of continuity


How do we quantify if a function is continuous, or has no jumps at a certain point, assuming the function is defined at that point? When a function has no jumps at point x = a, that means that when x is very close to a, f(x) is very close to f(a).

In other words, a function f is said to be continuous at a point, a, if for any arbitrarily small positive real number ε > 0 (ε is called epsilon), there exists a positive real δ > 0 (δ is called delta) such that whenever x is less than δ away from a, then f(x) is less than ε away from f(a), that is: |x - a| < δ guarantees that |f(x) - f(a)| < ε. For example, the following function is continuous at x = a:

Note how for any x in the interval (a - δ, a + δ), f(x) stays between the interval (f(a) - ε, f(a) + ε). Therefore, the above function is continuous at a.

The function below is not continuous because at x = a, if ε is less than the distance between the closed dot and the open dot, there is no δ > 0 for which the condition
|x - a| < δ guarantees |f(x) - f(a)| < ε. You will never find a delta such that all x satisfying |x - a| < δ also satisfy |f(x) - f(a)| < ε because the left part of the graph is disconnected from the right.