A function is differentiable at a point if it is continuous at that point and the derivative exists at that point. A function is considered continuous at a point if the limit of the function at that point exists and is equal to the function’s value at that point.ĭerivatives: Continuity is also an important concept when discussing the derivative of a function. Math topics that use Continuity Limits: In calculus, continuity is closely related to the concept of limits. We can plot the concentration of the chemical on the y-axis and the amount of chemical added on the x-axis to create a continuous curve. The concentration of a chemical in a solution: As a chemical is added to a solution, the concentration of the chemical in the solution will change. We can plot the velocity of the object on the y-axis and time on the x-axis to create a continuous curve. The velocity of a moving object: As an object moves, its velocity will change continuously as it accelerates or decelerates. We can plot the volume of the gas on the y-axis and the pressure and temperature on the x-axis to create a continuous curve. The volume of a gas in a container: As the pressure and temperature of a gas in a container change, the volume of the gas will also change. We can plot the height of the ball on the y-axis and time on the x-axis to create a continuous curve. The height of a ball as it bounces: When a ball bounces, its height will change continuously as it moves up and down. This graph will be a continuous curve, as the temperature of the substance will change smoothly and continuously over time. 5 real world examples of Continuity The temperature of a substance over time: As the temperature of a substance changes, we can plot its temperature on the y-axis and time on the x-axis. It was later refined and extended by other mathematicians, including Augustin-Louis Cauchy and Karl Weierstrass. The concept of continuity was first formally defined by the mathematician Bernard Bolzano in his work “The Paradoxes of the Infinite” in 1851. It’s important to remember that a function can be continuous at a point even if it is not differentiable at that point. One common mistake when working with continuity is to confuse it with other concepts, such as differentiability. In other words, there should be no sudden jumps or breaks in the function’s output values as the input values change.Ĭontinuity is typically introduced in calculus courses A function is said to be continuous at a particular point on its graph if the function’s value at that point is equal to its limit as the input value approaches that point. It involves understanding how a function’s output value changes as its input value changes. In Summary Continuity is a fundamental concept in calculus that deals with the smoothness and uninterrupted nature of functions. The question was worth two points.See Related Pages\(\) \(\bullet\text=1\) You could pull up the scoring guidelines to give students a real idea of what kind of justification is necessary. For consistency, refer to the definition of continuity in your text when discussing x-values not in the domain of a function.ĬYU #2 is directly from an old AP Calculus exam (2011, AB, 6a). Additionally, we call x=4 an infinite discontinuity, but some textbooks would not mention this point because it is not included in the domain of the function. ![]() In CYU #1, we intentionally included endpoints to generate conversation about this. Sharing the graph of a function with an oscillating discontinuity (like f(x) = sin (1/x)) may be valuable for students, but not tested on the exam.Īlthough it is not commonly tested, it is worth mentioning that a function can be continuous at its endpoint if the one-sided limit matches the y-value. Some teachers choose to include a third condition: f(a) exists, but we prefer to embed this condition within the shorter two-part definition. We recommend using the two-part definition of continuity (limits from left and right are equal and that the limit equals the y-value) and having students verify both conditions each time they must prove a function is continuous at a certain x-value. They will need more support in attaching the formal definition of continuity to their justifications. If they don’t have to pick up their pencil when tracing the graph from left to right, then the function is continuous. Students’ understanding of continuity is pretty intuitive. ![]() The debrief portion adds formal limit notation to each scenario. Students work through various cases and then infer the definition of continuity. There are many things that can go wrong that would cause the two people to not actually meet at Starbucks (i.e. This lesson introduces students to the idea of continuity using the analogy of a blind date.
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