Continuous functions which I must lift my pencil to draw

Question

Am I correct in thinking that the following three functions are continuous?

$$f(x)=\frac{(x-1)(x+2)}{x+2}$$

$$g(x)=\frac{x-1}{x+2}$$

Edit: because of my creativity with the graph, I'm going to have to share its definition for clarity: $$h(x)=\left\{\begin{array} &4.2 &1\leq x<3.75\\ 0.3x+4 & 3.75<x\leq 7 \end{array}\right.$$

Answer 1

Those are continuous functions, and that means they are continuous on their domains. However, their domains have limit points to which those functions cannot be extended continuously.

Answer 2

Yes, all three are continuous functions.

The definition of continuity at a point is based on the fact that the function has a value at that point (if a function is continuous at $x = a$, then $f(a)$ has a value in the expression $|f(x)-f(a)|<\epsilon$). However, it does not make sense to consider points at which a function is not defined. If we were asked to prove that the function is ‘discontinuous’ at a point, we would need to show that the condition for continuity at that point is false. And the negation of the condition for continuity ($\exists\epsilon >0\ \forall \delta>0,\ |x-a|<\delta \text{ and } |f(x)-f(a)|\geq\epsilon$) would not make sense at a point where the function value is not defined. In this way, I think that high school mathematics may often overlook this nuance in the definition of continuity. One usually thinks of continuity in a setting where a function is mapped from a set of real numbers, but let’s quickly extend this idea to a more abstract setting. If $X$ is a metric space, and $b$ is not an element of $X$, then is a function $f:X\to X$, that is otherwise continuous on $X$, discontinuous simply because it is not defined at $b$? One can clearly see the absurdity in this, and in the same way, we should only consider the subset of the real numbers that is the domain of a function when asking questions about continuity. The domains of the first two functions $f$ and $g$ above are $\mathbb R \setminus \{-2\}$.

For a textbook reference, see this page from M. Searcóid's textbook Metric Spaces.