Continuity: the Concept and Application of Continuity in Mathematics (Part I)

Continuity (Part I)

Continuity is the property of being continuous; that is, having no gaps or breaks. In mathematics, the concept of continuity has been generalized to include topological spaces, which are ways of classifying spaces according to their properties. This definition of continuity is used in mathematics and logic. The concept has applications in many fields, including philosophy and physics.

Limit of function in mathematics is defined as a value which a function approaches as the input approaches some value and the output remains within a specified number of decimal places.

The limit is considered to be infinite if the output goes beyond the specified number of decimal places. If there is no limit, it means that the function does not have any limits. It may take an infinite number of values or it may not have any value at all. If the output is approaching zero, then we can use l’hopital’s rule to find its limit.

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The Greatest Integer Function is a function which maps the natural numbers to the integers, and returns the greatest integer that is less than or equal to the input. It is defined as follows: GF(n) = gcd(n, n+1) – 1

Continuity is the property of a function whereby a curve can be drawn without lifting the pen from the paper, or in other words without any breaks.

Now let’s look at the formal definition of continuity, devise some continuity tests, and then discover some properties of continuous functions.

We can try to extend certain functions that are not continuous at one point to continuous functions.

Now, formally, we say that a function f(x) is continuous at an interior point c if f(c) exists and is equal to the value of f(x) at x = c.

We say that the function is continuous at the interior point c, meaning that if we consider the limit of the function as x approaches c, the limit exists and equals the value of the function at c.

When evaluating limits at the endpoints of an interval, we can consider left-hand and right-hand limits instead.

Let us look at the following graph. The blue line represents the graph of a function f(x). At the interior point c, it is continuous. This means that when we look at the limit of the function when x tends to c, it has a limit and it is exactly equal to the value of f(x) at c.

At the left-hand end point l, we only look at the right-hand limit when x tends to l, as there is no left-hand.

Thus, the function is continuous at the point l, meaning that the limit on the right side of f as x approaches l equals f(l). The situation with this boundary point here is similar, just that we only consider the left-hand limit instead of the right-hand limit.

Generally, a function is continuous on an interval if it is continuous at every point of the interval. A continuous function is a function that is continuous at every point in its domain of definition.

Moving to some examples, here is what we have.

Example:is the function f continuous at x=1

This graph shows a function given by the blue line, where it can be seen that we must lift our pen at least once to draw this line.

We see that the function is not continuous at the point x equals one, because it does not equal zero when x equals one. What gives us a clue that this is a breaking point?

We consider the limit of this function at the point x equals one. When x tends to one, the function has no limit and is therefore not continuous at this point.

The continuity fails here because there is no limit when x approaches one.

Second example now.

The graph is a continuous blue line. Can we say it is continuous?

Most likely, you’ll say that this is not continuous. The reason is that if you try to draw this graph, you cannot draw it in one stroke. There is a hole at x = 0.

To draw this graph, you need at least three separate strokes: to the left, to the right, and finally, set this point at x equals zero.

Again, there is a discontinuity at x equals zero. We see that the limit of the function at x equals zero exists. The limit equals zero. But it does not equal the value of the function at the point.

Thus, the graph fails to meet the condition for continuity.

Example:Locate the value of x for which the function is continuous

Consider the continuity of this function.

The graph of this function is not continuous. Where are the breaking points? You see clearly that if you try to draw the graph, you have to lift your pen from the paper at some point.

The first place to lift your pen is at point x equals zero.

Here, the function is not continuous, because the value of the function at the point x equals two is not equal to its limit. We are familiar with this.

Another breaking point is at x = 3, because the function does not have a limit when x approaches 3.

The function is continuous at all other places. The explanation for this continuity and discontinuity is given here.

Examine this.

The continuity test is used to determine if a function is continuous at a point. What would we call the criterion for continuity at the point x equals to c?

First, the value of the function at c must exist. Second, the limit of the function as x approaches c must exist. And thirdly, this limit must equal f(c).

This test has three parts, and the function is continuous at x equals c only if all three parts are satisfied. For example, in this graph you see clearly that blue line is a continuous function.

This graph is continuous at every point. You can draw it in one stroke. While many people believe that this graph has a breaking point, others believe that it is continuous throughout its entire length.

The function is not continuous at x = 1 because it does not have a limit as x approaches 1. We say that the function "jumps" at this point.

More generally, we have the greatest integer function, denoted by this symbol here, meaning given a real number x, this denotes the integer less than or equal to x.

The graph of this function is the horizontal line at zero between zero and one, the point one between one and two and the value two between two and three.

Now we call this function a step function and it has a discontinuity at every integer, like here, one, two, three, four and so on. This is actually a very useful and famous function in mathematics.

Here, we note that the left-hand limit at each integer n is equal to n - 1, while the right-hand limit at every integer n is basically equal to n itself.