In algebra, you study properties of a constant, a variable, or a function. A property of a constant is a particular value that does not change when the variable changes. For example, the value of x is set at some point, and it will stay the same throughout the function. A continuous function on the other hand is one which changes continuously and rapidly. For example, the rate of acceleration of a ball moving downhill is constant, but if it slows down and speeds up abruptly, that would be considered an abrupt change and it would need to be graphed out with a continuous function.
The concept of what is continuous in calculus can also be confusing because in many cases it is not. Take for example when working on the metric spaces. If you plot the distance between any two points on the metric space, what you get is a continuous function. However, it could also be a normal function that just goes left to right. The value of the metric space would then be a normal function of a continuous function.
What is continuity in calculus can be defined by a continuous function definition. It has a single defining value, and that value is the distance between any two points on the metric space. In many cases, this definition is called a ‘continuity’ since it is not changing as the function changes. So, for example, if we plot the line between x and y, we get a continuous function on the x-axis. If we plot the line between x and z, then we get a continuous function on the y-axis. A continuous function in this sense is actually just a function that repeats itself.
Another way to look at it is that there are two types of discontinuities in calculus: infinitesimals and discontinuous deformities. The first type are just different points on the map, while the second type are actual places on the map. The meaning of determining continuity in this case depends on what kind of infinitesimals or discontinuities are being used.
One example of a continuous function in the plane is the tangent line. The tangent line separates two points on the plane, hence its name, and it is a continuous function on both sides. The definition of the tangent line for continuity in calculus is ‘a function which separates zero values on one axis of a Cartesian coordinate system, from values on the other axis of the coordinate system’. If a tangent exists, then its value will always be zero on both planes. Thus, a continuous function is said to be constant on both sides. This is why you can always use the tangent plane as a reference point when working with graphs of constant functions.
The other example of a continuous function in calculus is the hyperbola. The hyperbola is defined as a tangent to a definite set of points (or on both sides), so it is a continuous function on both x and y. It follows naturally that the term ‘continuity in calculus’ refers to the meaning of the word ‘hyperbola’.
These two examples highlight the importance of understanding continuity in calculus. The trouble is, most students don’t understand the full spectrum of what is meant by continuity. Many don’t even realize that it’s an actual definition, and don’t think about ‘continuity’ in connection with calculus. But the better students realize the importance of continuity in the context of a broader philosophy of mathematics and learn to associate different terms such as ‘Hyperbola’ or ‘Cline’ with their own independent definition of continuity. In the final analysis, your knowledge of calculus is really only as good as the knowledge of the subjects you’re studying. And by learning to associate different terms (such as hyperbola and cline) with their own independent definition, you’ll be well on your way to mastering the subject and obtaining a thorough understanding of what is continuity in calculus.