Continuity and Differentiability - Intuitive definition of continuity

  • In mathematics, continuity is a fundamental concept that describes how a function behaves at every point within its domain.
  • A function is said to be continuous if it is possible to draw its graph without lifting the pencil from the paper.
  • Intuitively, a continuous function has no sudden jumps, breaks, or holes in its graph.

Definition of Continuity

  • A function f(x) is continuous at a point x = c if the following conditions are satisfied:
    1. f(c) is defined (i.e., c is in the domain of f).
    2. The limit as x approaches c of f(x) exists (i.e., the left-hand limit and right-hand limit of f(x) at x = c are both equal).
    3. The limit as x approaches c of f(x) is equal to f(c) (i.e., the value of the function at x = c equals the limit).

Examples

  1. The function f(x) = x^2 is continuous at all points on the real number line.
  1. The function g(x) = 1/x is continuous at all points except x = 0. The limit as x approaches 0 does not exist.

Properties of Continuous Functions

  • If f(x) and g(x) are continuous at x = c, then the following functions are also continuous at x = c:
    1. Sum of functions: f(x) + g(x)
    2. Difference of functions: f(x) - g(x)
    3. Product of functions: f(x) * g(x)
    4. Quotient of functions (if g(c) ≠ 0): f(x) / g(x)

Intermediate Value Theorem (IVT)

  • The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval [a, b] and takes on values f(a) and f(b), then it must also take on every value between f(a) and f(b) at least once.

Example

  • Let f(x) = x^3 - x - 1. Use the IVT to prove that f(x) has at least one root in the interval [1, 2].
  • We know that f(1) = 1^3 - 1 - 1 = -1 and f(2) = 2^3 - 2 - 1 = 3. Hence, f(x) takes values -1 and 3 at the endpoints of the interval [1, 2].
  • By IVT, f(x) must take on every value between -1 and 3 at least once, including zero. Therefore, f(x) has at least one root in the interval [1, 2].

Differentiability

  • Differentiability is a concept related to the rate at which a function changes with respect to its input.
  • A function f(x) is said to be differentiable at a point x = c if the following conditions are satisfied:
    1. f(c) is defined (i.e., c is in the domain of f).
    2. The limit as x approaches c of [f(x) - f(c)] / (x - c) exists and is finite.

Example

  • The function f(x) = x^2 is differentiable at all points on the real number line.
  • If we take the derivative of f(x) using the limit definition of the derivative, we get f’(x) = 2x, which exists for all x.

Relationship between Continuity and Differentiability

  • If a function f(x) is differentiable at a point x = c, then it must also be continuous at that point.
  • However, the converse is not always true. A function can be continuous at a point but not differentiable.

Continuity and Differentiability - Intuitive definition of continuity

  • A function f(x) is continuous at a point x = c if the following conditions are satisfied:
    1. f(c) is defined (i.e., c is in the domain of f).
    2. The limit as x approaches c of f(x) exists (i.e., the left-hand limit and right-hand limit of f(x) at x = c are both equal).
    3. The limit as x approaches c of f(x) is equal to f(c) (i.e., the value of the function at x = c equals the limit).

Examples

  • The function f(x) = x^2 is continuous at all points on the real number line.
  • The function g(x) = 1/x is continuous at all points except x = 0. The limit as x approaches 0 does not exist.

Properties of Continuous Functions

  • If f(x) and g(x) are continuous at x = c, then the following functions are also continuous at x = c:
    1. Sum of functions: f(x) + g(x)
    2. Difference of functions: f(x) - g(x)
    3. Product of functions: f(x) * g(x)
    4. Quotient of functions (if g(c) ≠ 0): f(x) / g(x)

Intermediate Value Theorem (IVT)

  • The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval [a, b] and takes on values f(a) and f(b), then it must also take on every value between f(a) and f(b) at least once.

Example

  • Let f(x) = x^3 - x - 1. Use the IVT to prove that f(x) has at least one root in the interval [1, 2].
  • We know that f(1) = 1^3 - 1 - 1 = -1 and f(2) = 2^3 - 2 - 1 = 3. Hence, f(x) takes values -1 and 3 at the endpoints of the interval [1, 2].
  • By IVT, f(x) must take on every value between -1 and 3 at least once, including zero. Therefore, f(x) has at least one root in the interval [1, 2].

Differentiability

  • Differentiability is a concept related to the rate at which a function changes with respect to its input.
  • A function f(x) is said to be differentiable at a point x = c if the following conditions are satisfied:
    1. f(c) is defined (i.e., c is in the domain of f).
    2. The limit as x approaches c of [f(x) - f(c)] / (x - c) exists and is finite.

Example

  • The function f(x) = x^2 is differentiable at all points on the real number line.
  • If we take the derivative of f(x) using the limit definition of the derivative, we get f’(x) = 2x, which exists for all x. `` Continuity and Differentiability - Intuitive definition of continuity

Slide 21:

  • A function f(x) is continuous at a point x = c if the following conditions are satisfied:
    1. f(c) is defined (i.e., c is in the domain of f).
    2. The limit as x approaches c of f(x) exists (i.e., the left-hand limit and right-hand limit of f(x) at x = c are both equal).
    3. The limit as x approaches c of f(x) is equal to f(c) (i.e., the value of the function at x = c equals the limit).

Slide 22:

Examples of Continuous Functions:

  • The function f(x) = x^2 is continuous at all points on the real number line because the above conditions are satisfied.
  • The function g(x) = 1/x is continuous at all points except x = 0 because the limit as x approaches 0 does not exist.

Slide 23:

Properties of Continuous Functions:

  • If f(x) and g(x) are continuous at x = c, then the following functions are also continuous at x = c:
    1. Sum of functions: f(x) + g(x)
    2. Difference of functions: f(x) - g(x)
    3. Product of functions: f(x) * g(x)
    4. Quotient of functions (if g(c) ≠ 0): f(x) / g(x)

Slide 24:

Intermediate Value Theorem (IVT):

  • The Intermediate Value Theorem states that if a function f(x) is continuous on a closed interval [a, b] and takes on values f(a) and f(b), then it must also take on every value between f(a) and f(b) at least once.

Slide 25:

Example: Use the IVT to prove the existence of a root.

  • Let f(x) = x^3 - x - 1. We want to prove that there exists at least one root in the interval [1, 2].
  • We know that f(1) = -1 and f(2) = 3. Hence, f(x) takes values -1 and 3 at the endpoints of the interval [1, 2].
  • By the IVT, f(x) must take on every value between -1 and 3 at least once, including zero. Therefore, f(x) has at least one root in the interval [1, 2].

Slide 26:

Differentiability:

  • Differentiability is a concept related to the rate at which a function changes with respect to its input.
  • A function f(x) is said to be differentiable at a point x = c if the following conditions are satisfied:
    1. f(c) is defined (i.e., c is in the domain of f).
    2. The limit as x approaches c of [f(x) - f(c)] / (x - c) exists and is finite.

Slide 27:

Example of a Differentiable Function:

  • The function f(x) = x^2 is differentiable at all points on the real number line. If we take the derivative of f(x), we get f’(x) = 2x, which exists for all x.

Slide 28:

Relationship between Continuity and Differentiability:

  • If a function f(x) is differentiable at a point x = c, then it must also be continuous at that point.
  • However, the converse is not always true. A function can be continuous at a point but not differentiable.

Slide 29:

Summary:

  • Continuity and differentiability are important concepts in calculus.
  • A function f(x) is continuous if it satisfies certain conditions at every point in its domain.
  • A function f(x) is differentiable if it has a defined rate of change at every point in its domain.

Slide 30:

  • Understanding the concepts of continuity and differentiability is crucial for solving problems in calculus and real-world applications.
  • By applying the properties of continuous and differentiable functions, we can analyze the behavior and properties of various mathematical models. ``