This function is not necessarily strictly monotonous, and y=f(x) has an inverse function, which only shows that f(x) is the so-called "injective function", that is, the corresponding law f is simply called simple mapping.

In short, injectivity means that different elements in the original image set correspond to different elements in the image set.

What is the mapping from 1 to 1? I think what he probably wants to express is a one-to-one mapping, which is both injective and injective. Functions with inverse functions are not necessarily surjective functions.

If a function is continuous, it must be strictly monotonous.

This function can represent some discrete points and a limited number of points in geometry. Who can say that these points must be monotonous?

Back to the eternal wanderer, I admit that the term "injective function" is not very standard, but I mention injective only to explain one-to-one mapping. I hope you can understand. However, this term is not useless. It was mentioned on page 125 of Discrete Mathematics (2007, China Water Resources and Hydropower Publishing House) edited by Jia Zhenhua.

Please take a look at the sixth page of Advanced Mathematics (Higher Education Press, 2002) edited by the Department of Applied Mathematics of Tongji University. It is clearly written that "only injectivity has inverse mapping", but are all injective functions monotonous? Obviously not. For example, f(x) is defined as follows: x= 1, 2,3,4 and f (1) =100f (2) =-25f (3) = 2589f (4) = 2365. This is an injective function. Obviously not.