It means, for a function to have continuity at a point, it shouldn't be broken at that point. A function is continuous at x a if and only if lim f(x) f(a). But how we translate this intuitive concept into the language of math, specifically, that of functions?Īs in many other instances, here graphs can help a lot to develop a clear intuitive understanding. Important Notes on Continuity: Here are some points to note related to the continuity of a function. Here we are talking about the physical intuition behind continuous functions. The opposite of something continuous (using this everyday usage of the word continuity) is called discrete. As we develop this idea for different types of intervals, it may be useful to keep in mind the intuitive idea that a function is continuous over an interval if we can use a pencil to trace the function. How many liters of water you drank yesterday. Now that we have explored the concept of continuity at a point, we extend that idea to continuity over an interval.The time it takes you to read this sentence.Īnd I hope you'll agree the following things are not continuous:.The change in velocity over time of an airplane taking off.The distance between a car and its destination (specifically, the change of distance).
For example, I'm pretty sure you'll agree that the following things are continuous: We know intuitively when something is continuous. It doesn't grow by leaps, but continuously. For example, the growth of a plant is continuous. In calculus, something being continuous has the same meaning as in everyday use. Now we can say that a function is continuous at a left endpoint of an interval if it is right continuous there, and a function is continuous at the right. In fact, calculus was born because there was a need to describe and study two things that we consider "continuous": change and motion. The concept of continuous functions appears everywhere.
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