Types of continuity

In this article, we’ll discuss limits and continuity. One of the most important and fundamental concepts in calculus is the idea of limits and continuity. A limit is a numerical digit that the function goes to when an independent function’s variable goes to a definite value. Another prominent topic in calculus is continuity. Examining if a pen could trace a graph of the function without taking the pen away from the page is a simple way to test for function continuity.

Limits

In Maths, a limit means the value approached by a series or function, as some value is approached by input or the index. Limits are important in calculus and are utilised to describe integrals, continuity and derivatives

Take y = f(x) as a function of x.

When at x = a, f(x) assumes an indeterminate form, we take into consideration the values of the function closer to a. If these values lean towards a definite unique number as x leans towards a, then the unique number, thus obtained, is called the limit of f(x) at x = a and we write it as x a lim ® f(x). 

L H L = x a lim  f(x) = L1 

Limit on the Left Hand If the value of f(x) can be set as near as needed to the digit II at a position close to a and on to the left of a, areal figure I) is indeed the left-hand side limit of  f(x) at x = a.

It could be written metaphorically as In other words, considering the values of f close to x to the left of a, x a lim f(x) is the estimated value of f at x = a. A left-hand side limit of function f(x) at a is referred to as this value.

Limitation on the right hand 

The right hand side limit of function f(x) at x = an is the areal number 12 if the values of f(x) could be set as near to the number 12 as desired at a position close to a and on to the right side of a. RHL = x a lim f(x) = l 1 may be represented symbolically as RHL = x a lim f(x) = l 1

In other words, given the values of f close to x to the right side of a, x a lim f(x) is the estimated value of f, at x = a. The right-hand side limit of f (x) at an is defined as this number.

Limits Existence If the right and left-hand limits are the same (i.e., they are the same), we may state that a limit exists, and also that common value is known as that of the limit of f(x) when x = a, and it is indicated by x a lim f. (x).

1. x 7 a 0 or x 7 a– is interpreted as x goes to a from the left and denotes that x is very near to a but always < a.
2. x a + 0 or x 7 a+ is interpreted as x seems to a from the right and denotes that x is extremely close to a but always larger than a.
3.  x 7 an is written as x goes to a and denotes that x is extremely close to but never equals to a.

Continuity

What is the mathematical definition of continuity?

In mathematics, a formal definition of the intuitive idea of a function that varies without sudden breaks or leaps is called continuity. A function is a connection in which each value of such an independent variable, such as x, is linked to a value of the dependent variable, such as y.

The geometrical relevance of continuity is that when a function is continuous, then its graphing doesn’t have breaks, but if it is discontinuous, the graph has. The point of discontinuity is the point whenever the graphs of the function break. A function is continuous at such a place if a limit exists there and is equal to the function’s value at that point. For example, a graph of functions sinx, x, and ex, among others, are continuous, but tan and sec x, among others, are discontinuous.

Continuity as defined by Cauchy At a point an in its domain, a f(x) is said to be continual if for every > 0, there exists a > 0 (depending on) such that |x – a| | f(x) – f(a) 1

Continuity as defined by Heine A function f is shown to be continuous at a point an of its domain D if the series f(a n) > converges to f(a) for any sequence < a n > of points in D converging to a, i.e. lim a n = a lim f (a n) = f (a)

Types of Continuity 

1. Continuity at one Point 

At x = a, a function f(x) is shown to be continuous if lim x a   f(a) = lim x a   f(x) = f(a), i.e. LHL = RHL = values of the f(x) at a.   We say f(x) is discontinuous at x = a  when it is not continuous at x = a.

2. In an Open Interval, Continuity If a function f(x) is continuous inside an open interval (a, b1), it is continuous at every point in (a, b), for example, y = [x] is continuous through (1,2).
3. In a Closed Interval, Continuity If a function f (x) is continuous inside a closed interval (a, b), it also seems to be continuous in that interval (a, b). 

Conclusion

We’ve discussed limits and continuity, their definition and formula and the types of continuity. Read through the article thoroughly to get a good grasp of the concepts mentioned and do as many practice problems on the topic as possible. Once the concept of limits and continuity is clearly understood, solving questions can become pretty easy.