Yes.

Publishing is a complicated process, especially in math, so such things happen from time to time. Authors usually put errata (a collection of known typos) on the book web pages if such pages exist.

Alec

I already said about that - there should be 2 things for continuity - the function should be defined, the limit should exist, and the value of the function should be equal to the limit.

In this case, the limit exists, but the function is not defined, so it is a discontinuity.

Maple has a function discont for finding discontinuities. It works well in this example,

discont((x^2-2)/(x-sqrt(2)),x);
                                  1/2
                                {2   }

If you do a plot, this point should be represented on it as an empty circle. Something like that,

with(plots): 
a := plot((x^2-2)/(x-sqrt(2))): 
b := pointplot([sqrt(2), 2*sqrt(2)+0.05], 
    symbolsize = 15, symbol = circle, color = blue): 
c := pointplot([sqrt(2), 2*sqrt(2)+0.05], 
    symbolsize = 10, symbol = solidcircle, color = white): 
display(a, b, c);

Alec

Limit exists, it is correct. In more detail, if you factor the numerator,

x^2-2 = (x - sqrt(2))*(x+sqrt(2))

and then cancel the (x-sqrt(2) term, you get x+sqrt(2). The function is equal to that in every point except x=sqrt(2). That's why the limit exists. It is not "undefined". And it can be seen from the plot.When x is approaching sqrt(2), y is approaching 2*sqrt(2) which is the limit.

The plot in Maple is constructed from points - in this example, 49 points, connected with line segments. The x-coordinates of points missed value x=sqrt(2), and there are no any problems in other points. So Maple connected those 49 points and got a straight line.

The correct plot should look as in the example I posted above.

Alec

1) I think, it is called removable, because the function can be made a continuous function after defining its value at that point being equal to the limit. For the other types of discontinuities that can not be done.

2) The jump discontinuities have the left and the right limits, but they are different. Infinite discontinuities have either left, or right limit infinite. "Oscillating" discontinuity doesn't have eiher left, or right limit. As one can see, all of them have some specific features. In many cases (but not all) the function is not defined at the discontinuity point. So it is usually good to check first. If a function is not defined - then it is a discontinuity and you don't even have to evaluate the limit to tell that. In particular, in your original example,

f:=x->(x^2-2)/(x-sqrt(2)):
f(sqrt(2));
Error, (in f) numeric exception: division by zero

One should be careful with that though, because Maple is not very good with that. See that thread, for example.

3) I think, it misses the singularity in most examples. It would be interesting actually to find an example in which the plot command would hit the discontinuity exactly, and see what happens. I don't know.

4) I tried to post a plot, but all the exports looked very bad. See that one, for example.

Also, there is a good SAGE tutorial for Calculus at that link. You might try it in addition (or instead) of Maple. SAGE developers are mathematicians, so Calculus in SAGE may be more mathematically correct than in Maple.

Alec