Continuity and Discontinuity

Continuity in mathematics means a function has no interruptions at a point, meeting three conditions: defined at the point, limit exists, and limit equals the function's value. Discontinuity occurs when these conditions fail, leading to point, jump, infinite, or oscillatory discontinuities. Understanding these concepts is essential for calculus, analyzing limits, integrability, and modeling real-world phenomena.

1.0Definition of Continuity

In mathematics, a function f(x) is continuous at a point x = a if the following conditions are met:

• f(a) is defined.
• The limit of f(x) as x approaches a exists: ${\lim }_{x\to a}f(x)$ .
• The limit equals the function's value at that point: ${\lim }_{x\to a}f(x)=f(a)$ .

If these conditions hold true for every point in its domain, the function is continuous throughout its domain.

2.0Definition of Discontinuity

In mathematics, a function f(x) is said to have a discontinuity at a point x = a if it fails to meet any of the conditions for continuity. Specifically, a discontinuity occurs if:

• f(a) is not defined.
• The limit of f(x) as x approaches a does not exist.
• The limit exists but is not equal to f(a).

Discontinuities can be classified into types such as point (removable), jump, infinite, or oscillatory discontinuities.

3.0Conditions for Continuity

In mathematics, a function f(x) is continuous at a point x = a if it satisfies the following conditions:

• Existence of the Function: f(a) must be defined.
• Existence of the Limit: The limit ${\lim }_{x\to a}f(x)$ must exist.
• Equality of the Limit and Function Value: The limit ${\lim }_{x\to a}f(x)$ must equal f(a).

If these conditions are met for every point x in the function's domain, f(x) is continuous over its entire domain.

4.0Continuity and Discontinuity of Functions

Continuity of Functions

A function f(x) is continuous at a point x = a if it satisfies the following three conditions:

1. Defined at x = a: f(a) exists.
2. Limit Exists: The limit ${\lim }_{x\to a}f(x)$ exists.
3. Limit Equals Function Value: The limit ${\lim }_{x\to a}f(x)$ equals f(a).

If a function meets these conditions at every point in its domain, it is continuous over that domain. Continuous functions do not have breaks, jumps, or holes in their graphs.

Discontinuity of Functions

A function f(x) has a discontinuity at a point x = a if it fails to satisfy any of the conditions for continuity. There are different types of discontinuities:

1. Point Discontinuity (Removable Discontinuity):

• The limit ${\lim }_{x\to a}f(x)$ exists, but f(a) is either not defined or f(a) $\ne {\lim }_{x\to a}f(x)$ .
• Example: $f(x)=\frac{\sin (x)}{x}$ for and f(0) = 1.

2. Jump Discontinuity:

• The left-hand limit ${\lim }_{x\to a}f(x)$ and the right-hand limit ${\lim }_{x\to a^{+}}f(x)$ exist but are not equal.
• Example: 

$f(x)=\{\begin{array}{ll}1& \text{ if }x<0\\ 2& \text{ if }x\ge 0\end{array}$

3. Infinite Discontinuity:

• The function approaches infinity near x = a, indicating a vertical asymptote.
• Example: $f(x)=\frac{1}{x}\text{ at }x=0$.

4. Oscillatory Discontinuity:

• The function exhibits erratic behavior near x = a, causing the limit to not exist.
• Example:  $f(x)=\sin \left(\frac{1}{x}\right)$ as x approaches 0.

5.0Continuous Function Properties

Continuous functions exhibit several important properties that make them a fundamental part of mathematical analysis. Here are some key properties:

1. Intermediate Value Theorem (IVT):

• If f(x) is continuous on the closed interval [a, b] and f(a) $\ne f(b$) , then for any value L between f(a) and f(b), there exists a point c in (a, b) such that  f(c) = L .
• This theorem ensures that continuous functions take on every value between f(a) and f(b).

2. Extreme Value Theorem (EVT):

• If f(x) is continuous on a closed interval [a, b], then f(x) attains both a maximum and a minimum value at some points within [a, b].
• This property guarantees that continuous functions on closed intervals are bounded and attain their extreme values.

3. Preservation of Intervals:

• If f(x) is continuous on an interval [a, b] and I is an interval contained in [a, b], then the image of I under f, denoted f(I), is also an interval.
• This property ensures that continuous functions map intervals to intervals, preserving the connectedness of the domain.

4. Additivity and Multiplicativity:

• If f(x) and g(x) are continuous at x = a, then f(x) + g(x) and f(x) $\cdot g(x)$ are also continuous at x = a.
• The sum and product of continuous functions are continuous.

5. Quotient of Continuous Functions:

• If f(x) and g(x) are continuous at x = a and g(a) $\ne 0$ , then the quotient $\frac{f(x)}{g(x)}$ is continuous at x = a.
• This property ensures that the division of two continuous functions is continuous, provided the denominator is non-zero.

6. Composition of Continuous Functions:

• If f is continuous at x = a and g is continuous at f(a), then the composite function 

$(\text{ gof })(x)=g(f(x))$  is continuous at x = a .

• This property allows the combination of continuous functions to produce a new continuous function.

7. Uniform Continuity:

• A function f(x) is uniformly continuous on an interval if, for any ε > 0, there exists a such that for all x, y in the interval, if |x-y|<\delta then |f(x)-f(y)|<0 $|\mathrm{f}(\mathrm{x})-\mathrm{f}(\mathrm{y})|<ϵ$
• Uniform continuity ensures that the function behaves well over the entire interval, not just at individual points.

6.0Continuity and Discontinuity Solved Examples

Example 1: Determine if the function f(x) = x² + 2x + 1 is continuous at x = 1.

Solution:

1. Check if f(1) is defined:

f(1) = 1² + 2(1) + 1 = 4

f(1) is defined and equals 4.

2. Find the limit as x approaches 1:

${\lim }_{x\to 1}\left(x^2+2x+1\right)$

As x approaches 1,

${\lim }_{x\to 1}\left(x^2+2x+1\right)=1^2+2(1)+1=4$

3. Verify if the limit equals f(1):

${\lim }_{x\to 1}f(x)=f(1)=4$

Since all three conditions are met, f(x) = x² + 2x + 1 is continuous at x = 1.

Example 2: Determine if the function$f(x)=\frac{x^2-1}{x-1}$ is continuous at x = 1.

Solution:

1. Check if f(1) is defined:

$\mathrm{f}(1)=\frac{1^2-1}{1-1}=\frac{0}{0}\text{ (undefined })$

Since f(1) is not defined, there is a point discontinuity at x = 1.

2. Simplify and find the limit:

Simplify the function:

$f(x)=\frac{(x-1)(x+1)}{x-1}=x+1(\text{ for }x\ne 1)$

3. Find the limit as x approaches 1:

${\lim }_{x\to 1}f(x)={\lim }_{x\to 1}(x+1)=1+1=2$

Although the limit exists and is 2, f(1) is undefined, confirming the point discontinuity.

Example 3: Determine if the function $f(x)=\{\begin{array}{ll}1& \text{ if }x<0\\ 2& \text{ if }x\ge 0\end{array}$ is continuous at x = 0.

Solution:

1. Check if f(0) is defined:   f(0) = 2

f(0) is defined and equals 2.

2. Find the left-hand limit:

${\lim }_{x\to 0^{-}}f(x)=1$

3. Find the right-hand limit:

${\lim }_{x\to 0^{+}}f(x)=2$

4. Compare the limits and f(0): 

${\lim }_{x\to 0^{-}}f(x)\ne {\lim }_{x\to 0^{+}}f(x)$

1 ≠ 2

Since the left-hand and right-hand limits are not equal, there is a jump discontinuity at x = 0.

Example 4: Determine if the function $f(x)=\frac{1}{x}$ is continuous at x = 0.

Solution:

1. Check if f(0) is defined:

$\mathrm{f}(0)=\frac{1}{0}(\text{ undefined })$

2.  Find the left-hand limit:

${\lim }_{x\to 0^{-}}f(x)={\lim }_{x\to 0^{-}}\frac{1}{x}=-\infty$

3. Find the right-hand limit:

${\lim }_{x\to 0^{+}}f(x)={\lim }_{x\to 0^{+}}\frac{1}{x}=\infty$

Since f(0) is undefined and the limits are infinite, there is an infinite discontinuity at x = 0.

7.0Continuity and Discontinuity Practice Problems

1. Determine if the following functions are continuous at the given points. If not, classify the discontinuity.

• $f(x)=\sqrt{x+1}\text{ at }x=0$
• $g(x)=\frac{\sin x}{x}\text{ at }x=0$

2. Identify and classify the point of discontinuity for the function: $h(x)=\frac{x^2-1}{x-1}$  
3. Identify and classify the jump discontinuity for the function: $k(x)=\{\begin{array}{ll}x& \text{ if }x<0\\ x+1& \text{ if }x\ge 0\end{array}$ 
4. Identify and classify the infinite discontinuity for the function: $\mathrm{m}(\mathrm{x})=\frac{1}{\mathrm{x}}$

8.0Solved Questions on on Continuity and Discontinuity

1. What is continuity in mathematics?

Ans: Continuity refers to the property of a function where there are no breaks, jumps, or holes. Formally, a function f(x) is continuous at a point x = a if ${\lim }_{x\to a}f(x)=f(a)$ .

2. How do you prove that a function is continuous at a point?

Ans: To prove continuity at a point x = a:

• Ensure f(a) is defined.
• Find ${\lim }_{x\to a}f(x)$ .
• Show that ${\lim }_{x\to a}f(x)=f(a)$ . 

3. What are the types of discontinuities?

Ans: Discontinuities in functions include:

• Point Discontinuity (Removable): f(a) is not defined or ${\lim }_{x\to a}f(x)$ does not exist but can be made continuous by defining f(a).
• Jump Discontinuity: Left-hand and right-hand limits exist but are not equal.
• Infinite Discontinuity:  ${\lim }_{x\to a}f(x)$ approaches infinity or negative infinity.
• Oscillatory Discontinuity: The function oscillates near x = a, preventing a limit from existing.

4. What are some properties of continuous functions?

Ans: Properties of continuous functions include:

• Preserves limits: ${\lim }_{x\to a}f(x)=f\left({\lim }_{x\to a}x\right)$ .
• Intermediate Value Theorem: If $\mathrm{f}(\mathrm{a})<\mathrm{L}<\mathrm{f}(\mathrm{b})$ , then there exists c in [a, b] where f(c) = L.
• Extreme Value Theorem: On a closed interval [a, b], f(x) attains both a maximum and minimum value.