HallsofIvy said:No. Did you have any reason at all for thinking that?
For [itex]x- y\ne 0[/itex]
[tex]\frac{x^3- y^3}{x- y}= x^2+ xy+ y^2[/itex]
As long as x is NOT equal to y, that function is the same as [itex]x^2+ xy+ y^2[/itex]. The function will be continuous on the line the line x= y, if we define [itex]f(x, y)= f(x, x)= x^2+ xy+ y^2= x^2+ x^2+ x^2= 3x^2[/itex], not "0".
Continuous extension is the process of extending or expanding a scientific theory or concept over time. It involves continuously building upon existing knowledge and making new discoveries to further understand a particular subject or phenomenon.
Continuous extension is crucial in scientific research because it allows scientists to continually improve upon and refine their understanding of a particular topic. It also helps to expand the scope of scientific knowledge and can lead to new breakthroughs and innovations.
There are many examples of continuous extension in scientific research, such as the development of new technologies, the discovery of new species, and the refinement of existing theories. For instance, the theory of evolution has undergone continuous extension as new evidence and research has been conducted.
Continuous extension is often confused with scientific methods such as experimentation and data analysis. However, continuous extension is a broader concept that encompasses these processes and refers to the ongoing growth and development of scientific knowledge over time.
One challenge of continuous extension is the potential for bias or preconceived notions to influence the interpretation of new data. Additionally, the rapid pace of technology and information can make it difficult for researchers to keep up with the latest advancements and incorporate them into their work.