Continuity at a Point

The concept of limits and continuity in calculus is one of the most important concepts to understand. Combinations of these notions have been extensively discussed in both Class 11 and Class 12, with many examples. A limit is defined as a number that is approached by the function as the independent function’s variable approaches a specified value in a mathematical equation. For example, if you have a function f(x) = 4x, you can state that “the limit of f(x) as x approaches 2 is 8.” This is a mathematical expression. It is written in the following way as a symbol:

lim x→2 (4𝑥) = 4×2=8

Continuity is another concept that is frequently discussed in calculus. The simplest method of determining whether or not a function is continuous is to evaluate whether or not a pen can trace the graph of a function without moving the pen off the paper. A conceptual definition is nearly always sufficient for studying precalculus and calculus, but a technical explanation is essential when studying higher-level calculus and statistics. By experimenting with limitations, you can discover a more accurate and precise method of determining continuity.

Definition of Continuity

Many functions have the advantage of allowing you to trace their graphs with a pencil without having to take the pencil away from the paper. They’re referred to as continuous functions. A function is said to be continuous at a particular point if its graph does not break there. A calculus basic course will, in general, provide a clear exposition of real function continuity in terms of the limit’s idea. First, a function f with variable x is continuous at the real line point “a” if the limit of f(x) as x approaches “a” is identical to the value of f(x) at “a,” i.e. f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = (a). Second, the function is continuous if it is continuous (as a whole) at every point in its domain.

Continuity can be mathematically described as follows:

If the following three conditions are met, a function is said to be continuous at a given point.

1. f(a) 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑

2.  f(x) 𝑒𝑥𝑖𝑠𝑡𝑠

3. f(x) = f(x)=f(a)

If you can trace the graph of a function without lifting the pen from the page, it is said to be continuous. A function, on the other hand, is said to be discontinuous if there are any gaps between its elements.

A continuous function’s graph is given below.

Discontinuity Types

Discontinuity can be divided into two categories:

• Infinite Discontinuity

• Jump Discontinuity

Infinite Discontinuity

The state of being unable to A branch of discontinuity with a vertical asymptote at x = a and no definition for f is referred to as infinite discontinuity (a). Another name for this is Asymptotic Discontinuity. If a function has values on both sides of an asymptote, it can’t be connected, hence it’s discontinuous at the asymptote. The graph below demonstrates this point.

Discontinuity in Jumps

A branch of discontinuity that consists of

f(x) ≠ f(x)

Both boundaries, on the other hand, are finite. This is also known as first-order continuities or simple discontinuities. Jump discontinuity is depicted graphically below.

Discontinuity in a Positive Way

A type of discontinuity in which a function has a two-sided limit at x = a, but f(x) is either undefined or not equal to that limit. Another word for this is a removable discontinuity.

This can be represented graphically as:

Definition of a Limit

The limit of a function is the number that it achieves when its independent variable reaches a specific value. The value (say a) to which the function f(x) approaches arbitrarily when the independent variable x approaches a given value “A,” indicated by f(x) = A.

Keep in mind the following:

• If  f(x)is the expected value of f at x = a given the values of ‘f’ near x to the left of a, then f(x) is the expected value of f at x = a. The left-hand limit of ‘f’ at an is known as this value

•  If  f(x) is the expected value of f at x = a given the values of ‘f’ near x to the right of a, then   f(x) is the expected value of f at x = a. The right-hand limit of f(x) at an is known as this value

• We call the common value the limit of f(x) at x = a and designate it by  f(x) if the right-hand and left-hand limits coincide (x)

Limitation on One Side

For an x-value that is slightly higher or lower than a specific value, the limit is totally defined by the values of a function. A two-sided constraint

lim x→a  f(x)

Both larger and smaller x values than an are taken into account. A one-sided restriction is seen from the left.

lim x→a–  f(x)

or from the appropriate side

 lim x→a– f(x)

only considers x values that are smaller or larger than a.

Limit Characteristics

As x tends to limit a, the limit of a function is denoted as f(x) reaches L.    f(x)=𝐿

The sum of two functions’ limits is equal to the sum of their limits, so:   f(x)+𝑔𝑋 = f(x)+ 𝑔𝑥

A constant term is the limit of any constant function, so  𝐶=𝐶

The product of the constant and the function’s limit is equal to the product of the constant and the function’s limit, resulting in:   𝑚 f(x)=𝑚 f(x)

[f(x)/g(x)] is a quotient rule. = lim x→a-f(a)/g(a)=lim x→af(a)/lim x→ag(a); if lim x→ag(a)0.

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 lim x→(LHL) ,and lim x→(RHL) are equal, 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.