Continuity over an interval

When studying "Continuity over an interval," here are the key points to focus on:

1. Definition of Continuity: A function $f(x)$ is continuous at a point $c$ if:

   • $f(c)$ is defined.
   • The limit of $f(x)$ as $x$ approaches $c$ exists.
   • The limit equals the function value: $\lim_{x \to c} f(x) = f(c)$.

2. Types of Continuity:

   • Continuous on an Interval: A function is continuous on an interval if it is continuous at every point in that interval.
   • Right and Left Continuity: Functions can be continuous from the right or left at a point, which helps in dealing with endpoints of intervals.

3. Types of Function Discontinuities:

   • Removable Discontinuity: A point where the function is undefined or has a hole, but could be made continuous by redefining $f(c)$.
   • Jump Discontinuity: Occurs when there is a sudden leap in function values.
   • Infinite Discontinuity: Happens when the function approaches infinity at a point.

4. Intermediate Value Theorem (IVT): If a function is continuous on a closed interval $[a, b]$ and takes values $f(a)$ and $f(b)$, then it also takes every value between $f(a)$ and $f(b)$ within that interval.

5. Continuous Functions: Common continuous functions include polynomials, rational functions (where the denominator is not zero), exponential functions, and sine and cosine functions.

6. Implication for Integrals and Limits: Continuous functions are easier to work with in calculus, particularly for evaluating integrals and finding limits.

7. Identifying Intervals of Continuity: Determine where a function is continuous by analyzing the definition and identifying any points of discontinuity.

By focusing on these key points, you will gain a fundamental understanding of continuity in the context of functions over intervals.

When studying "Functions Continuous on All Real Numbers," focus on the following key points:

1. Definition of Continuity:

   • A function $f(x)$ is continuous at a point $c$ if:
     • $f(c)$ is defined.
     • $\lim_{x \to c} f(x) = f(c)$.
     • The limit exists.

2. Types of Functions:

   • Polynomial Functions: Continuous everywhere.
   • Rational Functions: Continuous except where the denominator is zero.
   • Trigonometric Functions: Continuous on their domain.
   • Exponential and Logarithmic Functions: Continuous on their respective domains.

3. Intermediate Value Theorem:

   • If $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have different signs, there exists at least one $c \in (a, b)$ such that $f(c) = 0$.

4. Extreme Value Theorem:

   • A continuous function on a closed interval $[a, b]$ attains a maximum and minimum.

5. Compositions of Continuous Functions:

   • The composition of continuous functions is continuous.

6. Continuous Functions and Their Limits:

   • Continuous functions can be analyzed using limit properties.

7. Unbounded Behavior:

   • Understand how functions behave as they approach infinity (e.g., asymptotes).

8. Uniform Continuity:

   • Recognize that a function can be uniformly continuous on its domain, meaning continuity is maintained across all points in that domain.

9. Examples and Applications:

   • Analyze specific examples to understand how continuity manifests in different scenarios.

By focusing on these points, one can gain a solid understanding of functions continuous over all real numbers.

When studying "Functions Continuous at Specific x-Values," focus on these key points:

1. Definition of Continuity: A function $f(x)$ is continuous at a point $x = a$ if:

   • $f(a)$ is defined.
   • The limit of $f(x)$ as $x$ approaches $a$ exists.
   • The limit equals $f(a)$: $\lim_{x \to a} f(x) = f(a)$.

2. Types of Continuity:

   • Continuous from the Left: $\lim_{x \to a^-} f(x) = f(a)$
   • Continuous from the Right: $\lim_{x \to a^+} f(x) = f(a)$

3. Types of Functions: Recognize that:

   • Polynomial functions are continuous everywhere.
   • Rational functions are continuous except at points where the denominator is zero.
   • Trigonometric, exponential, and logarithmic functions have specific intervals of continuity.

4. The Squeeze Theorem: Useful for proving continuity at specific points, particularly when dealing with limits that are difficult to evaluate directly.

5. Discontinuity Types:

   • Removable: Limit exists but does not equal the function value.
   • Jump: Limits from the left and right do not match.
   • Infinite: The function approaches infinity at the point.
   • Oscillatory: The limit does not settle at a particular value.

6. Graphical Interpretation: Understanding the graphical representation of continuous functions versus discontinuous functions aids in visualization and comprehension.

7. Properties of Continuous Functions:

   • Continuous functions maintain their range and limits under operations such as addition, subtraction, multiplication, and composition.

8. Checking Continuity: To confirm continuity at $x = a$, check the three conditions systematically.

By mastering these points, you'll gain a solid foundation in understanding and identifying continuous functions at specific x-values.