Continuity in Functions

Study Notes

Continuity

A function (f(x)) is continuous at a point (x=a) if the following three conditions are met:
- (f(a)) is defined (i.e., the value of the function at (a) exists).
- (\lim_{x \to a} f(x)) exists (i.e., the limit of the function as (x) approaches (a) exists).
- (\lim_{x \to a} f(x) = f(a)) (i.e., the limit of the function as (x) approaches (a) is equal to the value of the function at (a)).
A function is continuous on an interval if it is continuous at every point in that interval.
Common types of discontinuities:
- Removable discontinuity: The limit exists, but the function is not defined or has a different value at the point.
- Jump discontinuity: The left-hand limit and the right-hand limit exist, but they are not equal.
- Infinite discontinuity: The function approaches infinity or negative infinity as (x) approaches the point.
- Oscillating discontinuity: The function oscillates infinitely as (x) approaches the point.

Differentiability

A function (f(x)) is differentiable at a point (x=a) if the derivative (f'(a)) exists.
The derivative (f'(a)) represents the instantaneous rate of change of the function at (x=a), and is defined as: (f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}).
A function that is differentiable at a point is also continuous at that point.
The converse is not necessarily true; a function can be continuous but not differentiable at a point. A classic example is the absolute value function at (x = 0).
Differentiability implies continuity.
Points where a function is not differentiable:
- Corners or cusps: The graph has a sharp turn. The tangent line is undefined.
- Vertical tangent: The graph has a vertical tangent line. The slope becomes infinite at that point.
- Points of discontinuity: The function is not continuous at the point.

Rules of Differentiation

Basic rules:
- Constant rule: (\frac{d}{dx} (c) = 0), where (c) is a constant.
- Power rule: (\frac{d}{dx} (x^n) = nx^{n-1})
- Sum/difference rule: (\frac{d}{dx} (f(x) \pm g(x)) = f'(x) \pm g'(x))
- Product rule: (\frac{d}{dx} (f(x) g(x)) = f'(x) g(x) + f(x) g'(x))
- Quotient rule: (\frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2})
- Chain rule: (\frac{d}{dx} (f(g(x))) = f'(g(x)) g'(x))

Applications of Derivatives

Finding tangents and normals to curves.
Finding maximum and minimum values of functions (optimization).
Determining intervals of increase and decrease.
Concavity and points of inflection.
Approximating values of functions using differentials.
Studying rates of change in various contexts (e.g., motion, growth).

Relationship between Continuity and Differentiability

Differentiability implies continuity.
Continuity does not imply differentiability.

Examples

Illustrative examples of finding the derivative of polynomial functions, trigonometric functions, and composite functions using rules.
Examples of finding maximum and minimum values of functions.
Examples of determining the nature of critical points.

Mean Value Theorem

A theorem that establishes a relationship between the average rate of change of a function over an interval and the instantaneous rate of change at some point within that interval.
Statement of the Mean Value Theorem.
Geometric interpretation of the Mean Value Theorem.

Higher Order Derivatives

The concept and calculation of higher order derivatives (second derivative, third derivative, etc.).
Applications of higher order derivatives (e.g., concavity, acceleration).

Description

This quiz focuses on the concept of continuity in mathematical functions. You will explore the conditions for a function to be continuous at a point and learn about different types of discontinuities. Test your understanding of limits and function behavior in this interactive quiz.