Introduction
I take it that the OP, now departed from the site, was exploring a pedagogical way to introduce the concept of continuity. I do not think they were trying to put continuity on a new rigorous basis. I once gave a talk, "Limits without delta," in which I described a middle ground between the "intuitive notion of a limit" and the "rigorous definition of a limit," as they are often called in US textbooks. The problem with learning a rigorous approach to limits is the algebraic handling of inequalities involving distance between numerical quantities. Students are comparatively okay at determining the sign of factored expressions. But distances involving the absolute value are an algebraic mess in their hands. They might manage the nested quantifiers — who knows? Others say no, but I claim the students' difficulties at coordinating the $\delta$ and $\epsilon$ inequalities
(not their fault, due to it not being taught at all)
makes testing that hypothesis impossible.
I intended the middle ground to give the students some practice at thinking with inequalities,
a skill that benefits from recurrent practice over a few semesters.
The idea started from Hardy's approach to sequences in A Course of Pure Mathematics and later got a boost while I was rereading Marsden and Tromba's Vector Calculus. In Marsden and Tromba, I noticed they kept using the squeeze theorem to manage $\epsilon$ without any need to solve for $\delta$. Another attack on $\epsilon-\delta$ is that it is more complicated than how mathematicians actually prove limits, and I wondered why we don't teach the squeeze theorem more effectively. In most textbooks, it resembles an appendix on the verge of appendicitis.
The definition below introduces continuity without reference to limits, in accordance with OP's course structure. It is nothing more than $\epsilon$-$\delta$ in disguise. An advantage is obtained by quickly introducing some squeeze theorems that the students can use in practical work.
Intuitive language based on the common notion of distance allow the students to avoid much of the algebra of inequalities that usually accompanies $\epsilon$-$\delta$. The work is not devoid of inequalities, but it is simplified. Again, the main goal is to understand the concept of continuity, and only in simplified contexts. In my mind the importance of continuity has its root source in our experience of motion and change in the world. Most of what we see or feel are continuous changes. Sudden changes — collisions, for instance — are imagined to be occur over a very small interval of time and not be "quantum leaps."
In class, I place
continuous change centered in the analysis of the change $f(x)-f(c)$ and emphasize that continuity is not just a kind of exam question but a model of change in the world.
Definition
The setup to be assumed:
- $c$ is a number;
- $f$ is a function;
- Some notion that $c$ is a "limit point" of the domain of $f$:
For first-time calculus students, let's say we will assume the domain of $f$ contains an open interval containing $c$. This is stronger than necessary but it keeps things simple.
$f$ is continuous at $c$ if for every positive distance, $f(x)$ lies within that distance of $f(c)$ for all $x$ that are sufficiently close to $c$ and belong to the domain of $f$.
Theorems and examples
Theorem 1. $f(x) = x$ is continuous at every real number.
Corollary 1.1. If you can find a positive constant $M$ such that $|f(x)-f(c)| \le M\cdot|x-c|$ for all $x$ sufficiently close to $c$, then $f$ is continuous at $c$.
Proofs: Theorem 1 is a tautology, more or less, and your best approach is not to try to prove what no one doubts. One might say, however close we would like $f(x)$ and $f(c)$ to be, well, that's the same as how close $x$ and $c$ would have to be, since $f(x)=x$ and $f(c)=c$. --
[Perhaps you would like to point out the meaning of being within some distance. Since $f(x)-f(c)$ is $x-c$, then what we need to show to prove continuity is that $|x-c|$ is small provided $|x-c|$ is small, a tautology. Even in formal $\epsilon$-$\delta$ proof, we quickly derive a tautology after letting $delta=\epsilon$.]
For the corollary, we have to divide by $M$ as follows: However close we would like $f(x)$ and $f(c)$ to be, say, less than some desired distance, such closeness is guaranteed whenever $x$ and $c$ are within $1/M$ times the desired distance. -- [If you want to introduce $\epsilon$-$\delta$, this is a good place to do it: Let $\epsilon$ be the distance that we desire $f(x)$ and $f(c)$ to be within. Then we would have $M\cdot|x-c| < \epsilon$ whenever $|x-c| < \epsilon/M$; etc.]
Discussion. For the corollary, warm-up proofs of special cases can be helpful. Take $f(x)=10x$. Then $f(x)-f(c)=10(x-c)$. So, if we want the distance between $f(x)$ and $f(c)$ to be less than $1/10$, how close to $c$ should $x$ be to be close enough? If the desired distance were $1/100$? Whatever distance I pick, how close would be close enough?
Likewise, we can show:
Corollary 1.2. If $g$ is continuous at $c$ and $|f(x)-f(c)| \le |g(x)-g(c)|$ for all $x$ sufficiently close to $c$, then $f$ is continuous at $c$.
With these tools it is as easy as calculating derivatives to show $\sqrt{x}$ and $x^n$ are continuous at $c \ne 0$. (The special case $c=0$ is easier but uses a different approach for roots.) The basic idea for $c\ne 0$ is to factor out $x-c$ from $f(x)-f(c)$ and bound the remaining factor. For well-chosen examples, finding the bound $M$ is a simple matter. For complicated examples, it is not. I avoid the complicated examples, since my goal is an understanding of the concept and not to train the students to be consummate provers of continuity. As for which examples are worth understanding, I would say the basic elements that make up functions expressed by formulas: powers, trig., log., and exp., as well as the ways functions are combined, $+$, $-$, $\times$, $\div$, and composition. One should end by remarking that a function given by a single formula in terms of elementary functions is continuous at each point in its domain.
Example from the OP.
$x^2$ is continuous at $2$.
Proof:
$x^2-2^2=(x+2)(x-2)$. If $x$ is closer to $2$ than one unit, then $1<x+2<3$. In that case, $|x^2-2^2| \le 5\cdot|x-2|$. By Corollary 1.1, $x^2$ is continuous at $2$.
Extension 1 of the example.
$x^2$ is continuous at all numbers $c$ in the interval $(1,3)$.
Proof:
Changing $5 = 3+2$ into $3+c$,
it's still the case that $|x^2-c^2| \le (3+c)\cdot|x-c| \le 6\cdot|x-c|$ for all $x$ in $(1,3)$.
Extension 2 of the example.
$x^2$ is continuous at all numbers $c$.
Proof:
Changing $5 = (2+1)+2$ into $(|c|+1)+|c|$,
it's still the case that $|x^2-c^2| \le (2|c|+1)\cdot|x-c|$ for all $x$ within $1$ unit of $c$.
Example 2.
The unit step function $u(x)$ is discontinuous at $0$.
Proof:
Over any open interval containing, the values of $u(x)$ are $0$ and $1$. So we cannot make $u(x)$ be closer to $u(0)$ than one unit for all $x$, no matter how close we require $x$ and $0$ to be.
Theorem 2.
If $f$ and $g$ are continuous at $c$, then so is $f+g$.
Proof:
$(f+g)(x)-(f+g)(c)=[f(x)-f(c)]+[g(x)-g(c)]$, so
$|(f+g)(x)-(f+g)(c)|\le|f(x)-f(c)|+|g(x)-g(c)|$.
By the continuity of $f$ and $g$ at $c$, each of $|f(x)-f(c)|$, $|g(x)-g(c)|$ may be made as small as we please. Thus their sum may be made as small as we please,
provided $x$ is sufficiently close to $c$. --
[If you feel the above is not pedantic enough:
Suppose we'd like $(f+g)(x)$ and $(f+g)(c)$ to be closer
than some desired distance.
Then in the first case, if $x$ is sufficiently close to $c$,
$|f(x)-f(c)|$ will be less than some part of it, say, half the deired distance. Likewise in the second case, if $x$ is sufficiently close to $c$,
$|g(x)-g(c)|$ will also be less than half the deired distance.
In the first case, $x$ has to be within one distance of $c$;
in the second, $x$ has be within a second distance of $c$.
Thus if $x$ is closer to $c$ than each distance,
then $|f(x)-f(c)|$ and $|g(x)-g(c)|$ will each be less than half the "desired distance." Therefore their sum will be less than the desired distance, and $|(f+g)(x)-(f+g)(c)|$ is at most their sum.]
Conclusion
One can then develop derivatives à la Stephen Kuhn (The Derivative à la Carathéodory. Am. Math. Monthly, 1991), which has conceptual advantages over the standard limit of the difference quotient; see this answer.
But the OP indicated that they would do limits next.
One warning: I find it easier to characterize various kinds of singularities with limits: discontinuities, nondifferentiable points, asymptotes. But getting across the main gist of calculus, which deals with smooth functions, seems simpler without limits.
Furthermore, students need to understand limits in subsequent courses, so one should not dispense with limits. They can be included without much trouble, including the characterizations of continuity and differentiability in terms of limits.
The upshot is that $\delta$ need only be found for $f(x)=x$, by the teacher, and for $f(x)=x^{1/n}$ in the case that $|x| < \delta \land \delta = \epsilon^n$ implies $|x^{1/n}| < \epsilon$, which the teacher can do. (For $n$ even, we would have to discuss one-sided continuity, which I've omitted from this answer.) The teacher might also elect to omit the $\epsilon$-$\delta$ proofs.
The approach frames continuity in terms of change in a (often physical) quantity, analyzes $f(x)-f(c)$ which analysis will be echoed in the subsequent discussion of differentiability, introduces inequalities related to the notion of "arbitrarily small," and cultivates "number sense."
The squeeze theorems teach the student to think in terms of bounds and estimate them. It feels satisfying when they realize that if $M=2$ is good enough to apply Corollary 1.1, then so is $M=100$.
Of course, there is something satisfying to the students when they can actually think with the concepts and not just push symbols around and calculate.
