Can you string together inequalities and equalities into a single statement?

If I write a statement like $$|f(x) - f(y)| \leq K|x - y| < K \delta = K \cdot \frac{\varepsilon}{K} = \varepsilon,$$ is that appropriate?

Context: I have a classmate who insists that "mixing" the equalities on the end of a statement that begins with inequalities somehow makes the statement inappropriate. I'm not sure if he thinks the above statement is literally wrong or (more likely) thinks it is somehow just stylistically unpleasant to the point where it should be avoided. My contention is that the statement is read from left to right and if it consists of a string of individually true statements, then the statement is true; and there isn't any other convenient way to write the statement, so it must be stylistically acceptable as well. Any rules, thoughts, or opinions?

I've seen and used that sort of "stringing" together quite frequently, and to my understanding it is well-understood what it means. There shouldn't be any issue with using it.

Whether it's "stylistically unpleasant" is more of a matter of opinion though. I have made some nasty such strings before, but they're usually rectified by just using line breaks, e.g.

$$\begin{align*} x &< y \\ &\le z \\ &= w \end{align*}$$

which is nice when $x,y,z,w$ are complicated expressions.

This is totally appropriate, and even recommended (at least to my opinion) to do when possible :)

This is pretty common and well-understood, but I think it's only understandable when it's immediately clear what relationship is being asserted for any two elements in the chain.

It also extends to other transitive predicate symbols like $\subset$.

For example, I think the following is pretty clearly a chain of inclusions.

$$ K \subset L \subset M $$

Piecing together what $x < y < z$ means from the meaning of $<$ alone is not possible. $x < y < z$ does not mean $(x < y) < z$ or $x < (y < z)$, neither of which are meaningful. Rather, it's equivalent to $(x < y) \land (y < z)$. This might be where your friend's objection comes from, especially if they've been exposed to things-similar-to-math-notation without this feature like most programming languages.

Relations without nice properties such as $\neq$ should probably be avoided.

$$ x \neq y \neq z $$

I would argue that is bad stylistically precisely because we can't tell whether $x \neq z$ is intended to be true or not.

Also, predicates that are transitive such as $|$ for divides that don't have an obvious orientation visually look odd when chained (at least to me).

$$ 2 | 4 | 60 $$

The correct term here is a chained relation. It's totally acceptable and common, and it's used throughout any math textbook I've ever seen. But this is a great question, because the form is never directly taught or assessed in any book or course, and therefore it's a perpetual source of confusion for students. At some point instructors start doing it, don't think of it as worth comment, and students may not even realize that it's happening or is important to note.

I actually find that I need to directly teach on the issue as a prerequisite to the discrete mathematics course that I run -- and I'm constantly referring students back to that document as they continue to make mistakes through the semester. I even recently wrote a blog article on the topic, the top three points being:

1. Even a single equality is hard for people to truly understand.
2. There is no explicit instruction in the form in any curriculum.
3. The compound form is entirely foreign to a natural language like English.

On that latter point, note there is continuing unresolved confusion about how to even pronounce out loud the simplest chained relation. For example, see questions on that topic separately at SE Math Educators and SE English Language -- which have entirely different top-voted answers!

Finally, I've found it necessary to even make an automatic timed quiz on reading chained relations for students on one of my websites. Interestingly, I've had different math-trained professionals tell me variously either (1) it's trivial and of no use, or (2) it's impossible given the span on the timer. So I think that signals a notorious "blind spot" in math education, in that it's never directly addressed, and thus very spotty if different people understand it intuitively or not.