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Limit and continuity in Calculus

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The limit of a function is an interesting but a little bit complex concept where, it may be possible that we try to find the value in the neighborhood of the Point of a function because the value of the function does not exist at a point. On the other hand, continuity of a function is closely related to the concept of limits. Now, talk about the definition of Limit, in many cases the values of the given function ‘f’ for the values of ‘x’ near a point ‘c’ and this value is not equal to function f (c) value or that value lies near no number at all.

Now, next concept to study is how we represent the limit, let a function 'f' is said to tends to a and the limit as 'x' tends to 'a', if x=a then it shows the values is just larger than or just smaller than x=a, f(x) has to move more closely to the value of limit and mathematically it can be written as,

Lim x→ a f(x) = 1, which is equal to

|f(x)-l| < e, x: 0 < |x-a| <d

Where, e and d are positive Numbers.

There are two types of limit that is Right and Left hand limits, in the Right Hand Limit

Lim x→ a+f(x) = 1, the value of a, is positive for function f(x).For example if the value of x tends to 1 than,

Lim x→ 1 (x) = 1,

And in Left Hand Limit the value of a, is negative for function f(x).The Mathematical Expression can be written as,

Lim x→ a-f(x) = 1,

For example if x tends to the value -1 than

Lim x→ 1-(x) = 0,

Now talk about infinity, infinity means something which keeps increasing and passes all limits this is called positive infinity. On the other side, something that continuously decrease and passes all limits is called negative infinity. The symbol of infinity is ∞. Some points given below which are necessary to read.

Infinity cannot be plotted on the paper.

∞+∞=∞

∞-∞ is indeterminate.

∞x∞=∞

0x∞=is indeterminate.

∞/∞,∞0 , are all indeterminate.

Now the definition of continuity, the any function that is f(x) is called continuous on an interval a if

Lim x→ a f(x) = f(a),

Otherwise, the function f(x) is discontinuous at a. Note that the continuity of f(x) at a means two things,

Lim x→ a f(x) the function exist

And this limit is f(a).

Lim x→ a f(x) = f(a)

This property is known as continuity for all Real Numbers a. The f(x) is said to be continuous from the left if the value of a is negative that is

Lim x→a- f(x) = f(a),

And the f(x) is said to be continuous from the right if the value of a is positive that is

Lim x→a+f(x) = f(a)

This shows the continuity and limits.

There are some points of continuity given below, if f(x) and g(x) are continuous at a, Then

(1) f(x) + g(x) is continuous.

(2) f(x) g(x) is continuous at a

(3) f(x)/g(x) is continuous at a g(a)≠0.

Continuity of function in Calculus

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A function is said to be continuous if function f ( x ) at Point ( c , f ( c ) ) if the conditions listed below are satisfied-

1) Function f ( c ) must exist.

2) The lim x → c f ( x ) must also exist.

3) And lim x → c f ( x ) = f ( c ).

The meaning of above conditions function is that there should be no missing point or gaps for f (x) at point ' c ' and we can move along the graph continuously. Function at point ( c , f (c ) ) is continuous from right if lim x -> c f ( x ) = f ( c ) and the function is continuous from left at ( c , f ( c ) ), if lim x → c f ( x ) = f ( c ).

If function does not follows above conditions or deviate from the above conditions then the function is discontinuous.

Let us take an example to understand the concept of continuity function in Calculus-

Example ) If f ( y ) = 2y + 3 at y = -4 then discuss the continuity of f ( x ).

Solution to the above problem is ) Using above 3 conditions we can discuss the continuity of the given function f ( y ). Therefore-

=> f ( - 4 ) = -5

=> lim y → - 4 f ( y ) = lim y → -4 ( 2y + 3 ) = -5

=> lim y → - 4 f ( y ) = f ( -4 )

Therefore we can say that the function f ( x ) = 2x + 3 is continuous at y = -4.

Continuity of function of one variable

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The continuity of a Functions can be determined by the fact that their graphs are continuous in nature i.e. the graph of the function is continuous. Continuous graphs are the type of the graphs in which there is no break in the graph and which are drawn without lifting the pencil from the paper. In the term of continuous Functions, it is also true that there is a sharp corner in the continuous functions. In the continuous graphs, there is a very small change in the dependent variable for a small change in an independent variable.

To understand continuity of a function, take a function y = f(x) and let x0 is a specific value of the function. Now, if the continuous process reflects to any other value x then the value of the function respect to that x will also reflects. The change in the value during this reflection is very small and this small difference (x - x0) will also gives the idea of a continuity of a function at a Point. In term of definition of continuity of function of one variable:

1. Function f(x) is defined.

2. Lim f(x) exists such that x -> x0 or

3. Lim(f(x)) = f(x0), x->x0

A continuous function approaches to two different limits according to the side that whether it is approaching to right or left. The graph of function f(x) such that where means x approaches towards ‘a’ from the right and means x approaches towards ‘a’ from the left. Function f(x) is said to approach b as a limit. The variable x approaches a only if the limit is b. It doesn’t matters that how x approaches to a.

Continuity of function of two or more variable

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A function f(x) is said to be continuous at x= a if limx→a- f(x) = limx→a+ f(x)=f(a).

i.e. if Left Hand Limit = Right hand limit = value of the function at 'a'.

i.e. limx→a f(x) = f(a),

If, f(x) is not continuous at x = a, we say that f(x) is discontinuous at x = a.

For a function to be continuous at any Point x=a, the function must be defined at that point and limiting values of f(x) when x approaches a, is equal to f(a).

Continuous function

Discontinuous function

Continuity of a function means the function should not break or there should not be any sudden jump at any point. A function 'f' of two variables is continuous at a point (x_0,y_0) if,

1. f(x_0,y_0) is defined

2. exists

3.

The above three requirements ensure that function f does not vary widely near the point and does not become infinite at the point, or have a jump discontinuity at the point. These all properties are the familiar properties of continuous Functions. Similar to the function of one variable the, functions of two or more variables are continuous on an interval if they are continuous at each point in the interval.

Example:

f (x, y) = (x + y )1/2,

D (f) = (x , y) : x + y > = 0.

This is an example of a continuous function, because the function remains the continuous on the whole Domain. The above is the brief introduction of the multivariable Functions with help of some examples.

Discontinuity of function in Calculus

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When we deal with Calculus we have to deal with Continuous Calculus and Discontinuous Calculus. In Continuous Calculus, we focus on continuous Functions and in discontinuous Calculus we deal with discontinuous Functions. Discontinuous functions are those functions which are not continuous means a function whose values does not vary continuously when the variable increases. For example, discontinuity by an abrupt change in the value of function, or discontinuity by an abrupt change in its law of variation, or when the function become imaginary, discontinuity occurs. Now, we discuss different type of discontinuity in function.

1. Discontinuity at certain Point: If function breaks at certain point, like f(x) = 1 /(x-1) here function 'f' is called as a discontinuous function at point x =2. Because at x=2 , f(x) tends to infinity so when we draw graph it breaks at x =2 discontinuity.

2. Discontinuity at many points: If function breaks at many points, like

f(x) = 2 when we factoring the denominator f(x) = 2
(x2 – x) x(x-1)

So, function breaks at two points x=0 and x=1.

These are some example which shows discontinuity of functions and for calculating, discontinuity in function we use differentiation at different limits like -if we want to calculate discontinuity of function f(x) at certain point x = a, then we have to calculate differentiation at x=a-h where h is very small quantity and we calculate differentiation at x= a+h and if both differentiation are equal, then this function f(x) is discontinuous at x= a.

lim(x->a-h) d(f(x) = lim(x->a+h) d(f(x) here d(f(x) means differentiation of f(x), lim(x->a-h) shows limit at x= a-h and lim(x->a+h) shows limit at x= a+

Types of Discontinuity in Calculus functions

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Continuous Functions are those Functions which have no breaks i.e. when we draw them on a graph it seems to be a smooth curve or graph from one end to the other end without any cut or break between it. To understand Discontinuity we need to learn different types of discontinuous functions.

Continuous at x=a.

If a function is not continuous function than it is said to be discontinuous function, this means it have cut or breaks in its graph.

Discontinuous at x = a.

After brief introduction of Discontinuous Functions, move to different types of Discontinuous Functions. Let g(x) = log x since, here limx→0 + g(x) exists whereas limx→0 + g(x) does not exists, because negative values of x are not defined for log x.

  1. Removable discontinuous functions: If limx→b g(x) exists but it is not equal to the function g(b), then the function g(x) has removable discontinuity at x = b and by redefining g(x) for x = b we can remove it. Let’s take an example to understand it:

Example: Redefine the function g(x) = [x] + [-x] in such a way that it becomes continuous for x ∈(0, 2).

Solution:

Here limx→1 g(x) = -1 but g(b)=0

Hence, g(x) has a removable discontinuity at x = 1.

To remove this we define g(x) as follows

g(x) = [x] + [-x], x ∈(0, 1) U (1, 2)

= -1, x = 1.

Now, g(x) is continuous for x ∈(0, 2)

2. Non-removable discontinuous functions:

If limx→b g(x) does not exists, then we can remove this discontinuity from the function so that this function become a non-removable or essential discontinuous function. Let’s take an example to understand it:

Example: g(x) = [x + 3] has essential discontinuity at any x ∊1.

3. Jump discontinuous functions:

A function g(x) can be said discontinuous at a Point x = b if, limx→b - g(x) ≠ limx→b + g(x) and g(x) and may be equal to either of previous limits. Let’s take an example to understand it:

Example: g(x) = [ x ] : [ -x ]. This expression denotes that greatest Integer have jump discontinuity at all integer values.

Take an example of discontinuous function:

Question 1. Find the points of discontinuity of the function f(x) = [ (1 - x2, x < 0) and (x + 2, x >= 0 ) ] if they exist.


Solution: Here this function exists for all value of x, since it is not elementary because it is defined by two different functions. We will analyse the behaviour of the function near to the point x = 0 where it changes its analytic expression.

Calculate one-sided limits at x = 0.

limx->0-0 f(x) = limx->0-0 (1 - x2) = 1,

limx->0+0 f(x) = limx->0+0 (x + 2) = 2.

Thus, this function has a discontinuity of the first kind in it at x = 0. The finite jump at x = 0 is:

Δy = limx->0+0 f(x) - limx->0-0 f(x) = 2 – 1 = 1.

This function is continuous for all other values of x, because both the functions are defined from the left and right of the point x = 0 are elementary functions without any discontinuities.