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4 min read•february 15, 2024
Now that we’ve explored different types of discontinuity, we can dive into what continuity really is and how we can define it at a point. 🎯
Continuity means that a function behaves smoothly and doesn't have sudden jumps or gaps in its graph. It's like drawing a line without lifting your pen. To check if a function is continuous, we make sure it's not broken or full of holes, and it behaves predictably as we get closer to specific points. This concept is important, especially in calculus and real analysis, to understand how functions change and interact.
On the AP exam, you’re going to have to be able to justify WHY something is continuous (or not continuous). Let's explore ways to figure out if a function is continuous!
A function f(x) is continuous at a specific point 'c' in its domain if the following three conditions are met:
1️⃣ f(c) is defined (i.e., there is a value of the function at c)
2️⃣ The limit of the function as x approaches c exists
3️⃣ The value of the function at c (f(c)) is equal to the limit of the function as x approaches c. In other words, .
To prove that a line is continuous at a specific point using a graph, you'll need to ensure that there are no jumps, gaps, or breaks in the graph at that point. The key is to visually demonstrate that the line flows smoothly without any interruptions. Here's how you can do it:
For example, take a look at these graphs:
In this example, only the top left graph is continuous. On all other graphs, there is a skip which makes the graph discontinuous at that point. Below the graphs are explanations as to why the graph is not continuous. These explanations are what we should aim for!
Is the function continuous at ? Justify your conclusion using the definition.
Explanation: To check continuity, we need to ensure three things:
Consider the functions and . Are both functions continuous at ? Justify your conclusions using the definition.
Explanation: For each function, apply the definition of continuity:
Perfect, both functions are continuous at x = 3. ✏️
Examine the functions and . Are both functions continuous at ? Justify your answer.
Explanation:
Therefore, function q(x) is continuous at x = 0, while p(x) is not continuous at x = 0.
Good luck, you got this! 🍀
|x|
: The absolute value of a number x is its distance from zero on a number line. It always returns a non-negative value.Complex Functions
: Complex functions are functions that take complex numbers as inputs and produce complex numbers as outputs. Complex numbers consist of both a real part and an imaginary part.Continuity
: Continuity describes whether or not there are any breaks, holes, or jumps in a function. A continuous function has no interruptions and can be drawn without lifting your pen from the paper.Defined at a Point
: When we say that a function is defined at a point, it means that there is an actual value assigned to that specific input within the domain.Function
: A relationship between two sets where each input (domain) value corresponds to exactly one output (range) value.Limit of the Function
: The limit of a function represents the value that the function approaches as the input approaches a certain point or infinity.Mathematical Notation
: Mathematical notation refers to the symbols, symbols, and conventions used to represent mathematical ideas and relationships. It allows mathematicians to communicate complex concepts concisely.Real Functions
: Real functions are functions that take real numbers as inputs and produce real numbers as outputs. They describe relationships between real-world quantities such as time, distance, temperature, or money.sin(1/x)
: The function sin(1/x) is an oscillating function that varies rapidly as x approaches zero. It has infinitely many peaks and valleys, but it never settles on a single value.sqrt(x)
: The square root of a number x is the value that, when multiplied by itself, equals x. It gives the positive solution to the equation x = y^2.x^2*e^x
: The expression x^2*e^x represents a function that combines the power of x squared with the exponential growth of e^x. It is a common example of an algebraic function with both polynomial and exponential terms.4 min read•february 15, 2024
Now that we’ve explored different types of discontinuity, we can dive into what continuity really is and how we can define it at a point. 🎯
Continuity means that a function behaves smoothly and doesn't have sudden jumps or gaps in its graph. It's like drawing a line without lifting your pen. To check if a function is continuous, we make sure it's not broken or full of holes, and it behaves predictably as we get closer to specific points. This concept is important, especially in calculus and real analysis, to understand how functions change and interact.
On the AP exam, you’re going to have to be able to justify WHY something is continuous (or not continuous). Let's explore ways to figure out if a function is continuous!
A function f(x) is continuous at a specific point 'c' in its domain if the following three conditions are met:
1️⃣ f(c) is defined (i.e., there is a value of the function at c)
2️⃣ The limit of the function as x approaches c exists
3️⃣ The value of the function at c (f(c)) is equal to the limit of the function as x approaches c. In other words, .
To prove that a line is continuous at a specific point using a graph, you'll need to ensure that there are no jumps, gaps, or breaks in the graph at that point. The key is to visually demonstrate that the line flows smoothly without any interruptions. Here's how you can do it:
For example, take a look at these graphs:
In this example, only the top left graph is continuous. On all other graphs, there is a skip which makes the graph discontinuous at that point. Below the graphs are explanations as to why the graph is not continuous. These explanations are what we should aim for!
Is the function continuous at ? Justify your conclusion using the definition.
Explanation: To check continuity, we need to ensure three things:
Consider the functions and . Are both functions continuous at ? Justify your conclusions using the definition.
Explanation: For each function, apply the definition of continuity:
Perfect, both functions are continuous at x = 3. ✏️
Examine the functions and . Are both functions continuous at ? Justify your answer.
Explanation:
Therefore, function q(x) is continuous at x = 0, while p(x) is not continuous at x = 0.
Good luck, you got this! 🍀
|x|
: The absolute value of a number x is its distance from zero on a number line. It always returns a non-negative value.Complex Functions
: Complex functions are functions that take complex numbers as inputs and produce complex numbers as outputs. Complex numbers consist of both a real part and an imaginary part.Continuity
: Continuity describes whether or not there are any breaks, holes, or jumps in a function. A continuous function has no interruptions and can be drawn without lifting your pen from the paper.Defined at a Point
: When we say that a function is defined at a point, it means that there is an actual value assigned to that specific input within the domain.Function
: A relationship between two sets where each input (domain) value corresponds to exactly one output (range) value.Limit of the Function
: The limit of a function represents the value that the function approaches as the input approaches a certain point or infinity.Mathematical Notation
: Mathematical notation refers to the symbols, symbols, and conventions used to represent mathematical ideas and relationships. It allows mathematicians to communicate complex concepts concisely.Real Functions
: Real functions are functions that take real numbers as inputs and produce real numbers as outputs. They describe relationships between real-world quantities such as time, distance, temperature, or money.sin(1/x)
: The function sin(1/x) is an oscillating function that varies rapidly as x approaches zero. It has infinitely many peaks and valleys, but it never settles on a single value.sqrt(x)
: The square root of a number x is the value that, when multiplied by itself, equals x. It gives the positive solution to the equation x = y^2.x^2*e^x
: The expression x^2*e^x represents a function that combines the power of x squared with the exponential growth of e^x. It is a common example of an algebraic function with both polynomial and exponential terms.© 2024 Fiveable Inc. All rights reserved.
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