Continuity at a point

When studying "Continuity at a point," focus on these key points:

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

   • $f(c)$ is defined.
   • $\lim_{x \to c} f(x)$ exists.
   • $\lim_{x \to c} f(x) = f(c)$.

2. Limit Existence: Understand that the limit must approach the same value from both the left and right sides as $x$ approaches $c$.

3. Types of Discontinuities:

   • Removable Discontinuity: The limit exists, but $f(c)$ is either undefined or not equal to the limit.
   • Jump Discontinuity: The left-hand limit and right-hand limit exist but are not equal.
   • Infinite Discontinuity: The limit approaches infinity or negative infinity.

4. Graphical Interpretation: Be able to analyze and sketch graphs to visualize continuity and identify points of discontinuity.

5. Continuous Functions: Familiarize yourself with properties of continuous functions (e.g., composition, addition, multiplication) and how they behave over intervals.

6. Intermediate Value Theorem: Understand the implications of continuity on the behavior of functions, such as guaranteeing that functions take on every value between f(a) and f(b) for $a < b$.

Studying these points will give you a comprehensive understanding of continuity at a point in calculus.

When studying "Worked example: Continuity at a point (graphical)," focus on these key points:

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

   • $f(c)$ is defined.
   • $\lim_{x \to c} f(x)$ exists.
   • $\lim_{x \to c} f(x) = f(c)$.

2. Graphical Interpretation: Look for whether the graph has any breaks, jumps, or holes at the point $x = c$.

3. Left-hand and Right-hand Limits: Ensure both limits from the left and right are equal and match the function's value at $x = c$.

4. Types of Discontinuities:

   • Removable: A hole exists in the graph.
   • Jump: There’s a sudden change in function value.
   • Infinite: The function approaches infinity or negative infinity.

5. Visual Analysis: Practice sketching graphs and identifying points of continuity by observing the behavior near the point in question.

By mastering these points, you can effectively analyze and determine the continuity of functions at specific points.

When studying the topic of pointwise continuity of a function, here are the key points to focus on:

1. Definition of Continuity: Understand that a function $f(x)$ is continuous at a point $c$ if the following three conditions are met:

   • $f(c)$ 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 Limits: Familiarize yourself with different types of limits, including one-sided limits (left-hand limit and right-hand limit).

3. Evaluating Limits: Learn various techniques to evaluate limits, such as direct substitution, factoring, rationalizing, and applying L'Hôpital's Rule if necessary.

4. Discontinuities: Recognize the types of discontinuities:

   • Removable discontinuities (holes).
   • Jump discontinuities (a sudden jump in function value).
   • Infinite discontinuities (asymptotes).

5. Piecewise Functions: Understand how to analyze piecewise functions for continuity by checking the conditions at the boundaries of each piece.

6. Graphical Interpretation: Be able to interpret continuity visually by analyzing function graphs for breaks or abrupt changes at particular points.

7. Examples: Study worked examples that illustrate how to determine the continuity of functions at specific points, including both polynomial and rational functions.

By focusing on these key points, you can gain a solid understanding of pointwise continuity in functions.

When studying the topic of points where a function isn't continuous, key points to focus on include:

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

   • $f(c)$ is defined.
   • $\lim_{x \to c} f(x)$ exists.
   • $\lim_{x \to c} f(x) = f(c)$.

2. Types of Discontinuities:

   • Removable Discontinuity: Occurs when a function's limit exists at a point but doesn't equal the function's value there.
   • Jump Discontinuity: Occurs when the left-hand limit and right-hand limit at a point are different.
   • Infinite Discontinuity: Occurs when the function approaches infinity as it nears the point.

3. Identifying Discontinuities:

   • Evaluate the function around the point in question.
   • Check limits from both sides and compare to the function's value at the point.

4. Graphical Interpretation: Understanding how discontinuities appear on a graph helps in visualizing and identifying them.

5. Examples: Analyze specific functions to see how and where they are not continuous, applying the definitions and types mentioned.

By focusing on these points, you'll have a solid understanding of where and why functions may fail to be continuous.