Continuity is a mathematical concept that describes the smoothness and connectedness of a function. It means that a function has no abrupt changes or breaks, and can be drawn without lifting the pen from the paper.
Proving continuity is important because it helps us understand the behavior of a function and make accurate predictions about its values. It also allows us to use powerful mathematical tools, such as the Intermediate Value Theorem and the Mean Value Theorem.
To prove continuity, we need to show that the function is defined at a particular point, the limit of the function at that point exists, and the value of the function at that point is equal to the limit. This can be done using the formal definition of continuity or by using the properties of continuous functions.
Some common techniques for proving continuity include using the definition of continuity, using the properties of continuous functions such as the sum, difference, product, and quotient rules, and using the Squeeze Theorem. We can also use graphical and numerical methods to support our proof.
Some functions fail to be continuous because they have breaks or abrupt changes in their graphs, or they have a point of discontinuity where the limit of the function does not exist. This can happen due to removable discontinuities, jump discontinuities, or infinite discontinuities.