Continuity and Discontinuity in Calculus

Continuity and Discontinuity: Continuity and discontinuity are fundamental concepts in calculus and mathematical analysis, describing the behavior of functions. A function is continuous at a point if you can draw the graph of the function at that point without lifting your pen from the paper. Continuity implies that small changes in the input of the function result in small changes in the output, making the function predictable and smooth.

A function is discontinuous at a point x = c if it fails to be continuous at that point. In this article, we will discuss the Continuity and Discontinuity of functions with their conditions and types.

Continuity Definition

A function is said to be continuous if one can sketch its curve on a graph without lifting the pen even once. A function is said to be continuous at x = a, if, and only if the three following conditions are satisfied.

Conditions for Continuity

1. The function is defined at x = a; that is, f(a) equals a real number
2. The limit of the function as x approaches a exists
3. The limit of the function as x approaches a is equal to the function value at x = a

A function f(x) is said to be continuous in the open interval (a, b) if at any point in the given interval the function is continuous. In the case of closed interval [a, b], the function is said to be continuous:

• f(x) is be continuous in the open interval (a, b)
• lim_x⇢a f(x) = f(a)
• lim_x⇢b f(x) = f(b)

Example 1: Prove that the function f(x) = 5x - 3 is continuous at x = 0.

Solution:

Given, f(x) = 5x - 3

At x = 0 , f(0) = (5 × 0) - 3 = -3 
lim_x⇢0 f(x) = lim_x⇢0 (5x - 3) = (5 × 0) - 3 = -3 
lim_x⇢0 f(x) = f(0) 

Therefore, f(x) is continuous at x = 0.   

Example 2: Examine the function f(x) = |x - 5|, for continuity.

Solution:

Given function, f(x) = |x - 5| 

Domain of f(x) is real and infinite for all real x 
Here , f(x) = |x - 5| is a modulus function 
As , every modulus function is continuous 
Therefore , f(x) is continuous in its domain R.   

Example 3: Is the function f(x) =  x - sinx + 5  is continuous at x = π.

Solution:

Given function is f(x) = x - sinx + 5 

L.H.L = lim_x⇢π- (x - sinx + 5) = lim_x⇢π- [(π - h) - sin(π - h) + 5] = π + 5 

R.H.L = lim_x⇢π+ (x - sinx + 5) = lim_x⇢π+ [(π + h) - sin(π + h) + 5] = π + 5 

And, f(π) = π - sinπ + 5 = π  + 5 

Since , L.H.L = R.H.L = f(π) 
Therefore , f(x) is continuous at x = π 

Example 4: Examine the continuity of the function f(x) = 2x - 1 at x = 3.

Solution:

Given f(x) = 2x - 1 

At x = 3, f(x) = (2 × 3) - 1 = 5 
lim_x⇢3 f(x) = lim_x⇢3 f(x) = (2×3) - 1 = 5 
lim_x⇢3 f(x) = f(3) 
Therefore, f(x) is continuous at x = 3

Example 5: Examine the function is continuous or not?

Solution:

For x > 0, y = x and x < 0, y = -x

So, We Know it is continuous for x > 0 and x < 0. To check if it is continuous at x = 0 , check the limit:

lim_x⇢0-  |x| = lim_x⇢0- (-x) = 0

lim_x⇢0+  |x| = lim_x⇢0+ (x) = 0

So, lim_x⇢0 |x| = 0 , which is equal to the value of the function at 0. Therefore, It is continuous everywhere. 

Discontinuity Definition

A function is discontinuous at a point x = a if the function is not continuous at a. The function "f" is said to be discontinuous at x = a in any of the following cases:

1. f(a) is not defined
2. lim_x⇢a+ f(x) and lim_x⇢a-  f(x) exists, but are not equal.
3. lim_x⇢a+ f(x) and lim_x⇢a- f(x) exists and are equal but not equal to f(a).

Types of Discontinuity

There are three basic types of discontinuities

1. Removable(point) Discontinuity
2. Infinite Discontinuity
3. Jump Discontinuity

Removable(point) Discontinuity: The graph has a hole at a single x-value. Imagine you're walking down the road, and someone has removed a manhole cover. This is a category of discontinuity in which the function has a well defined two-sided limit at x = a, but either f(a) is not defined or f(a) is not equal to its limit.

• lim_x⇢a f(x) ≠ f(a)
• f(a) = lim_x⇢a f(x)

Infinite Discontinuity: The function goes toward positive or negative infinity. Imagine a road getting closer and closer to a river with no bridge to the other side. The function diverges at x = a to give it a discontinuous nature here. That is to say, f(a) is not defined. Since the value of the function at x = a tends to infinity or doesn’t approach a particular finite value, the limits of the function as x → a are also not defined.

Jump Discontinuity: The graph jumps from one place to another. Imagine a superhero going for a walk, he reaches a dead end and, because he can, flies to another road. In this type of discontinuity, the right-hand limit and the left-hand limit for the function at x = a exists; but the two are not equal to each other.

• lim_x⇢a+  f(x) ≠ lim_x⇢a- f(x)

Example 1: Find all the points of discontinuity of the function f defined by f(x) = |x| - |x+1|.

Given function = |x| - |x+1|

From the function the critical points are x=0 and x=-1

For x< -1 , f(x) = -x - (-x-1) = 1

For -1<=x<0 , f(x) = -x-(x+1) = -2x-1

For x>= 0 , f(x) = x - (x+1) = -1

Checking the Discontinuity :

At x = -1

Left limit: lim_x⇢-1-f(x) = 1

Right limit: lim_x⇢-1+ f(x) = -2(-1)-1 = 1

Function value: f(-1) = 1

Continuous at x = -1 (limits and function value are equal )

At x = 0

Left limit: lim_x⇢-0-f(x) = -2(0) -1 = -1

Right limit: lim_x⇢-0+ f(x) = -1

Function value: f(0) = -1

Continuous at x = 0 (limits and function value are equal )

Since the function is continuous at x = -1 and x = 0, there are no points of discontinuity. Thus, f(x) is continuous everywhere.

Example 2: Show that the function defined by g (x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x

For x approaching an integer n from the left (x→ n^- )

lim _x⇢n- g(x) = 1

because [x] = n - 1

For x approaching n from the right (x→n⁺)

lim _x⇢n+ g(x) = 0

because [x] = n

Since the left-hand limit and the right-hand limit are not equal at any integer n, g(x) is discontinuous at all integers.