Continuity over an interval

When studying "Continuity over an interval," 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.
   • The limit of $f(x)$ as $x$ approaches $c$ exists.
   • The limit of $f(x)$ as $x$ approaches $c$ equals $f(c)$.

2. Types of Discontinuities:

   • Point Discontinuity: The limit exists, but does not equal the function’s value.
   • Jump Discontinuity: The limits from the left and right at a point exist but are not equal.
   • Infinite Discontinuity: The function approaches infinity at a point.

3. Continuity on Intervals: A function is continuous over an interval if it is continuous at every point within that interval (open, closed, or half-open).

4. Properties of Continuous Functions:

   • Continuous functions on closed intervals achieve both maximum and minimum values (Extreme Value Theorem).
   • The Intermediate Value Theorem states if $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, there exists at least one $c \in (a, b)$ such that $f(c) = N$.

5. Common Continuous Functions: Polynomial, exponential, and trigonometric functions are generally continuous everywhere.

6. Discontinuous Functions: Recognize functions that may be piecewise or have defined points where they are not continuous.

7. Graphical Interpretation: Understand how continuity can be interpreted graphically, with a continuous function appearing as an unbroken curve.

By mastering these points, you will have a solid foundation in the concept of continuity over intervals in calculus.

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.
   • The limit $\lim_{x \to c} f(x)$ exists.
   • $\lim_{x \to c} f(x) = f(c)$.

2. Types of Continuous Functions:

   • Polynomial Functions: Always continuous.
   • Rational Functions: Continuous except where the denominator is zero.
   • Trigonometric Functions: Generally continuous, with specific points to note (e.g., points of discontinuity in tangent).
   • Exponential and Logarithmic Functions: Continuous in their respective domains.

3. Properties of Continuous Functions:

   • The sum, difference, product, and quotient (where defined) of continuous functions are also continuous.
   • Continuous functions on closed intervals are bounded and reach their maximum and minimum values (Extreme Value Theorem).

4. Intermediate Value Theorem: If $f$ is continuous on [a,b] and $N$ is any value between $f(a)$ and $f(b)$, there exists at least one $c \in (a,b)$ such that $f(c) = N$.

5. Uniform Continuity: A function is uniformly continuous if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x, y$ in the domain, if $|x - y| < \delta$, then $|f(x) - f(y)| < \epsilon$.

6. Continuous Functions on the Real Line: Functions that maintain continuity over all real numbers may have certain behaviors like oscillation, but must adhere to the overall properties and definitions stated above.

By mastering these points, you will have a solid understanding of functions continuous on all real numbers.

When studying functions that are continuous at specific x-values, 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.
   • The limit of $f(x)$ as $x$ approaches $c$ exists.
   • The limit of $f(x)$ as $x$ approaches $c$ equals $f(c)$.

2. Types of Discontinuities:

   • Removable Discontinuity: Occurs when the limit exists but does not equal the function value.
   • Jump Discontinuity: Occurs when the limits from the left and right at $c$ exist but are not equal.
   • Infinite Discontinuity: Occurs when the function approaches infinity or negative infinity as $x$ approaches $c$.

3. Calculating Limits: Use various techniques (substitution, factoring, etc.) to compute limits at the specific x-values of interest.

4. Piecewise Functions: For functions defined in pieces, check the continuity at the boundaries of the pieces.

5. Intermediate Value Theorem: For continuous functions, the theorem states that if $f$ is continuous on the interval $[a, b]$, then it takes every value between $f(a)$ and $f(b)$.

6. Continuity in Composite Functions: The composition of continuous functions is also continuous.

7. Continuous Functions on Intervals: A function can be continuous over a range and not continuous at isolated points.

8. Graphical Interpretation: Understanding the continuity of functions often requires analyzing their graphs for breaks, jumps, or holes.

By mastering these concepts, you will be well-equipped to analyze continuity at specific points in functions.