Derivatives - Relation between Continuity and Differentiability

• In this topic, we will explore the relationship between continuity and differentiability of a function.

Continuity

• A function is said to be continuous at a point if its graph has no breaks, jumps, or holes at that point.
• For a function to be continuous at a point, three conditions must hold:
  1. The function must be defined at that point.
  2. The limit of the function at that point must exist.
  3. The limit of the function at that point must be equal to the value of the function at that point.

Differentiability

• A function is said to be differentiable at a point if it has a derivative at that point.
• Geometrically, this means that the graph of the function has a well-defined tangent line at that point.
• The derivative can be interpreted as the rate of change of the function at that point.

Relationship between Continuity and Differentiability

• If a function is differentiable at a point, it is also continuous at that point.
• However, the converse is not true. A function can be continuous at a point without being differentiable at that point.
• In other words, differentiability implies continuity, but continuity does not imply differentiability.

Example 1

Consider the function: - f(x) = |x|

• This function is continuous everywhere, but it is not differentiable at x = 0.
• The function has a sharp corner at x = 0, which prevents the existence of a well-defined tangent line at that point.

Example 2

Consider the function: - f(x) = x^2 sin(1/x)

• This function is continuous everywhere, but it is not differentiable at x = 0.
• As x approaches 0, the function oscillates infinitely between -1 and 1, preventing the existence of a well-defined tangent line at that point.

Equation for Differentiability

• A function f(x) is differentiable at a point c if and only if the derivative of f(x) exists at that point.
• The derivative, denoted by f’(c), can be calculated using the limit definition of the derivative: f’(c) = lim(h -> 0) [f(c+h) - f(c)] / h

Equations for Continuity

• A function f(x) is continuous at a point c if and only if the following conditions hold:
  1. f(c) is defined (i.e., f(c) exists).
  2. lim(x -> c) f(x) exists.
  3. lim(x -> c) f(x) = f(c).
• These conditions ensure that there are no jumps, breaks, or holes in the graph of the function at the point c.

Summary

• Continuous functions do not have any breaks or jumps in their graphs.
• Differentiable functions have well-defined tangent lines at each point.
• Differentiability implies continuity, but continuity does not imply differentiability.
• The derivative of a function at a point is a measure of its rate of change at that point.
• The limit definition of the derivative and the conditions for continuity provide mathematical tools to analyze the behavior of functions.

Limit Definition of the Derivative

• The derivative of a function f(x) at a point c can be defined using the limit definition:
  • f’(c) = lim(h -> 0) [f(c + h) - f(c)] / h
• This formula calculates the rate of change of the function at the point c.
• The limit ensures that the difference quotient becomes a well-defined value.
• It measures the instantaneous rate of change or slope of the function at that specific point.

Differentiability Implies Continuity

• If a function is differentiable at a point c, it must also be continuous at that point.
• This means that the function has no breaks, jumps, or holes in its graph at the point c.
• The derivative measures the rate of change, so for it to exist, the function needs to be smooth and continuous.
• However, being continuous does not guarantee differentiability.
• A function can be continuous but not have a well-defined derivative at certain points.

Example: Differentiability and Continuity

• Consider the function:
  • f(x) = x^3
• This function is differentiable and continuous everywhere.
• The derivative of the function is:
  • f’(x) = 3x^2
• The derivative exists and is well-defined for all values of x.
• This example demonstrates that differentiability implies continuity.

Example: Continuity Without Differentiability

• Consider the function:
  • f(x) = |x|
• This function is continuous for all values of x, as it has no jumps or breaks in its graph.
• However, the function is not differentiable at x = 0.
• The derivative does not exist at x = 0 because of the sharp corner in the graph of the function.
• This example shows that continuity does not imply differentiability.

Differentiability and the Tangent Line

• The derivative of a function at a point c represents the slope of the tangent line to the graph at that point.
• A tangent line is a straight line that touches the graph at only one point and has the same slope as the graph at that point.
• If a function is differentiable at a point, the tangent line exists and is well-defined at that point.
• The equation of the tangent line can be determined using the point-slope form: y - y₁ = m(x - x₁), where m is the slope.

Example: Tangent Line

• Consider the function:
  • f(x) = x^2
• The derivative of this function is:
  • f’(x) = 2x
• Let’s find the tangent line to the graph of f(x) at the point (2, 4):

  • Slope, m = f’(2) = 2(2) = 4

  • Using the point-slope form, the equation of the tangent line is: y - 4 = 4(x - 2)

• The equation simplifies to: y - 4 = 4x - 8
• This example shows how the tangent line represents the rate of change (slope) of the function at a specific point.

Summary: Continuity and Differentiability

• Continuity and differentiability are closely related concepts in calculus.
• A function is continuous if its graph has no breaks or jumps, and differentiable if the tangent line exists at each point.
• Differentiability implies continuity, but continuity does not guarantee differentiability.
• The derivative measures the rate of change or slope of a function at a specific point.
• The limit definition of the derivative provides a formula to calculate the derivative of a function at a point.

Further Analysis of Functions

• The concepts of continuity and differentiability have broad applications in mathematical analysis.
• These concepts help us understand the behavior and properties of functions on different intervals.
• By studying the continuity and differentiability of a function, we can determine its local and global extrema, inflection points, and critical points.
• The derivative provides valuable information about the shape and behavior of a function near a specific point.
• It also helps in curve sketching and optimization problems.

Applications of Continuity and Differentiability

• The concepts of continuity and differentiability are not only important in calculus but also have wide-ranging applications across various fields.
• Physics: Continuity and differentiability play a crucial role in describing motion, forces, and physical phenomena using mathematical equations.
• Engineering: These concepts are used in designing and analyzing systems, optimization problems, and modeling real-world processes.
• Economics: Continuity and differentiability are employed in economic modeling, analyzing supply and demand curves, and optimizing production functions.
• Computer Science: These concepts are essential in algorithms, numerical methods, and data analysis.

Review Questions

1. Can a function be differentiable without being continuous?

2. Is a function continuous if its derivative exists at every point?

3. Explain the relationship between continuity and differentiability.

4. What does the derivative measure? Give an example.

5. How can the limit definition of the derivative be used to calculate the derivative at a point?

Differentiability at a Point

• A function f(x) is differentiable at a point c if and only if the derivative of f(x) exists at that point.
• The derivative can be interpreted as the instantaneous rate of change of the function at that point.
• The derivative can also be calculated using the derivative rules for different functions, such as power rule, product rule, quotient rule, and chain rule.
• If a function is differentiable at a point, it means that the function is smooth and has a well-defined tangent line at that point.
• The tangent line represents the rate of change or slope of the function at the specific point.

Example: Differentiability at a Point

• Consider the function:
  • f(x) = 2x^2 - 3x + 1
• The derivative of this function is:
  • f’(x) = 4x - 3
• Let’s find the value of the derivative at x = 2:
  • f’(2) = 4(2) - 3 = 8 - 3 = 5
• Since the derivative exists at x = 2, the function f(x) is differentiable at that point.

Differentiability on an Interval

• A function may be differentiable at some points and not at others.
• If a function is differentiable at every point within an interval, it is said to be differentiable on that interval.
• Differentiability on an interval implies continuity on that interval, as continuity is a necessary condition for differentiability.
• The differentiability of a function on an interval can be analyzed by checking the differentiability of its component functions and applying relevant derivative rules.

Example: Differentiability on an Interval

• Consider the function:
  • f(x) = |x|
• The function f(x) is continuous everywhere, but it is not differentiable at x = 0.
• Therefore, f(x) is differentiable on the intervals (-∞, 0) and (0, ∞), but it is not differentiable at x = 0.

Differentiability and Discontinuity

• If a function has a discontinuity at a point, it cannot be differentiable at that point.
• Discontinuities include jump discontinuities, removable discontinuities (holes), and essential discontinuities.
• At a jump discontinuity, the function has a sudden jump or break in its graph, preventing the existence of a well-defined tangent line.
• At a removable discontinuity, the function has a hole in its graph, which again prevents the existence of a well-defined tangent line.
• At an essential discontinuity, the function exhibits oscillation or divergence, making the derivative undefined.

Example: Discontinuity and Differentiability

• Consider the function:
  • f(x) = 1/x
• This function has a removable discontinuity (hole) at x = 0.
• Since the function is not defined at x = 0, it cannot be differentiable at that point.
• However, the function is differentiable at all other points.

Differentiability and Lipschitz Continuity

• Lipschitz continuity is a stronger form of continuity that implies differentiability.
• A function f(x) is said to be Lipschitz continuous if there exists a constant K such that:
  • |f(x) - f(y)| ≤ K|x - y| for all x and y in the domain of the function.
• If a function is Lipschitz continuous, it is automatically continuous.
• Differentiability is guaranteed for Lipschitz continuous functions except possibly at isolated points.

Example: Lipschitz Continuous Function

• Consider the function:
  • f(x) = √x
• This function is Lipschitz continuous on any closed interval [a, b], where a and b are positive numbers.
• The derivative of the function is:
  • f’(x) = 1/(2√x)
• The derivative exists and is well-defined for x > 0, which covers the entire domain of the function.
• Therefore, f(x) is differentiable on any closed interval [a, b], where a and b are positive numbers.

Summary

• Differentiability is a property of a function that indicates the existence of a well-defined derivative.
• A function is differentiable at a point if the derivative exists at that point.
• Differentiability implies continuity, but continuity does not guarantee differentiability.
• Discontinuities in a function prevent differentiability at those points.
• Lipschitz continuity is a stronger form of continuity that guarantees differentiability except possibly at isolated points.
• The derivative provides valuable information about the behavior, rate of change, and slope of a function at specific points.

Review Questions

1. Explain the difference between differentiability at a point and differentiability on an interval.

2. Can a function be differentiable at a point but not continuous at that point? Provide an example.

3. How do different types of discontinuities affect differentiability?

4. What is Lipschitz continuity, and how does it relate to differentiability?

5. What information does the derivative provide about a function?