Overview of Continuity on an Interval

Continuity is defined at a single location; a function’s limit must exist at that location; and the value of the function at that location must equal the value of the limit at that location in order for the function to be continuous at that location. Detachable discontinuities, jump discontinuities, and endless discontinuities are the three forms of discontinuities. When a function is continuous at every point in an open interval, it is referred to as being continuous over the interval in which it occurs. Continuity over a closed interval is achieved when the interval is continuous at every point in its interior and at its ends.

 In many functions, the ability to trace their graphs with a pencil without lifting the pencil from the page is a significant advantage over other functions. They’re referred to as continuous functions in this context. Other functions have points where the graph breaks, but they only meet this property across intervals within their respective domains, not across the entire domain. They are continuous during these periods, and they have a discontinuity where a break is experienced.

As a first step in our investigation of continuity, we consider what it means for a function to be continuous at a specific point in time. On the surface, a function is considered continuous at a particular point if the graph of the function does not break at that point.

Continuity at a Crossroads

 Consider some functions that do not correspond to our intuitive sense of what it means for a function to be continuous at some time before considering what it means for a function to be continuous at some point in the future. Then we create a set of factors that will prevent such failures from occurring in the future.

 The initial function of interest is depicted in the figure. We can see that there is a hole in the f(x) graph at the point a. In reality, f(a) is an undefined function. The following criterion must be met in order for f(x) to be continuous at point an at the very least:

 f(a) has been defined.

• limxaf(x) exists.

• limxaf(x)=f(a).

Discontinuities of Various Types

 Discontinuities can manifest themselves in a variety of ways, as demonstrated in Examples 1 and 2. Disposable discontinuities, infinite discontinuities, and jump discontinuities are the three types of discontinuities we’ve encountered so far in our exploration of discontinuities. Detachable discontinuities are those that have a hole in the graph, leap discontinuities are those that occur when the sections of the function do not meet, and infinite discontinuities are those that occur at the vertical asymptote of the function. The figure illustrates the differences between the various types of discontinuities. Even while these terminologies are useful for distinguishing between three different types of discontinuities, it is important to remember that not all discontinuities fall into one of these classifications.

Continuity across a Time Period

Since we’ve examined the concept of continuity at a single moment, we’ll now consider the concept of continuity over time. In order to expand this idea for different kinds of intervals, it may be good to recall that a function is continuous over an interval if we can trace it with a pencil between any two points in the interval without raising the pencil from the paper. In order to define continuity on an interval, we must first consider what it means for a function to be continuous from the right at a point and continuous from the left at a point in order to understand what it means to be continuous from the left at a point.

Continuity from the Right and from the Left

A function  f(x)  is said to be continuous from the right at a if limxa+f(x) =f(a).  

A function  f(x)  is said to be continuous from the left at a if  limxa-f(x) =f(a).  

When a function is continuous at every point in an open interval, it is referred to as being continuous over the interval in which it occurs. It is said to be continuous across a closed interval of type [a,b] if it is continuous at all points in (a,b) and is continuous from the right at a and left at b, as well as being continuous at all points in the closed interval of type [a,b]. The same is true for functions that are continuous over intervals of the form (a,b] and are continuous from the left at b, if they are continuous throughout the interval. Continuity is defined in the same way for other types of intervals as it is for the first.

We can trace the graph of the function from point (a,f(a)) to point (b,f(b)) without lifting the pencil since limxa+f(x) =f(a). and limxb-f(x) =f(b). are required. If we had limxa+f(x) f(a).Instead, we’d have to raise our pencil to go from f(a) to the graph of the rest of the function over (a,b).

The Theorem of Intermediate Values

It is possible to find a variety of advantageous characteristics in functions that are continuous over intervals corresponding to the pattern [A,B], where a and B are real numbers. There are numerous important theorems about such functions that we will come across throughout our calculus education. The Intermediate Value Theorem is the first of these theorems, and it is the most important.

For simplicity, consider that f is continuous across the interval [a,b], which is both closed and bounded. If z is any real number between f(a) and f(b), then there is an integer c in [a,b] that fulfils the condition f(c)=z in Figure (b).

Important Concepts

• Continuity is defined at a single location; a function’s limit must exist at that location; and the value of the function at that location must equal the value of the limit at that location in order for the function to be continuous at that location.

• Detachable discontinuities, jump discontinuities, and endless discontinuities are the three types of discontinuities that exist.

• When a function is continuous at every point in an open interval, it is referred to as being continuous over the interval in which it occurs. Continuity over a closed interval is achieved when the interval is continuous at every point in its interior and at its ends.

• If f(x) is continuous at L and limxag(x) =L., then

  limxaf(g(x)) = f(limxag(x)) =L, according to the composite function theorem.

• The Intermediate Value Theorem ensures that a function takes on every value between its ends if it is continuous across a closed interval..

Conclusion 

Limits and continuity are key notions in calculus. Class 11 and 12 have extensively discussed these concepts with examples. A limit is a number that the independent function’s variable approaches in a mathematical equation.

Continuity implies that if the left hand, and the function value at x=a exist and are identical, the function f is continuous at x=a. Discontinuous functions are undefined or nonexistent.