3
$\begingroup$

Yesterday, when my lecturer was reading course lecture, she said that this function $$f(x)=\frac{x^2-x-2}{x-2}$$ has a breakpoint and is not continuous, also if we sketch graph it will have jumping point at $x=2$, it is clear from his equation because $x-2$ at $x=2$ has a vertical asymptote, but if we factorize the given equation as $\frac{(x-2)(x+1)}{(x-2)}$ then we could cancel out $x-2$, yes? (as I know in this case $x=2$ is removable point or something like this). If we cancel out, then we are left only with $x+1$, which is continuous everywhere. I have also tried to use wolframalpha to see the graph, here it is

enter image description here

It shows me the graph of straight line, so my question is: was my teacher's belief that this function is discontinuous correct or am I wrong?

$\endgroup$
5
  • 8
    $\begingroup$ It is not continuous at $2$ because it is not defined at $2$. But as the picture shows, the discontinuity is removable. The function $g(x)$, defined as above for $x\ne 2$, and as $3$ at $2$, or more simply as $x+1$ everywhere, is continuous. There is no vertical asymptote for $f(x)$. $\endgroup$
    – André Nicolas
    Nov 26, 2011 at 7:37
  • 2
    $\begingroup$ I can't tell if your teacher said this or you said it, but the function does not have a vertical asymptote at $x=2$. Vertical asymptote refers specifically to the "exploding" behavior you see in a function like $\frac{1}{x}$ near $x = 0$. As your graph demonstrates, this behavior does not take place near $x = 2$. $\endgroup$
    – Austin Mohr
    Nov 26, 2011 at 7:53
  • 2
    $\begingroup$ vertical asymptote i have added,not my teacher,she said only discontinuity thanks $\endgroup$
    – dato datuashvili
    Nov 26, 2011 at 7:54
  • 1
    $\begingroup$ A more interesting related question in classical analysis: When in general is a discontinuity for a real valued map removable? A rational function with a factorable polynomial in the numerator with a factor in the denominator is an almost trivial case-but when is it true in general?In complex analysis,for holomorphic functions,there's a very well defined theory based on a theorem of Riemann. But what about real valued maps? Hmmmmmmm............. $\endgroup$
    – Mathemagician1234
    Nov 26, 2011 at 8:03
  • 1
    $\begingroup$ it is interesting $\endgroup$
    – dato datuashvili
    Nov 26, 2011 at 8:06

3 Answers 3

Reset to default
7
$\begingroup$

At $x=2$ it is not defined, and

$$\lim_{x \rightarrow 2} \frac{x^2-x-2}{x-2}=\lim_{x \rightarrow 2} \frac{(1+x)(x-2)}{x-2}=\lim_{x \rightarrow 2} (1+x)=3$$

Because the limit exists, the discontinuity is removable.

$\endgroup$
1
  • $\begingroup$ thanks @Daniel and all guys $\endgroup$
    – dato datuashvili
    Nov 26, 2011 at 8:26
4
$\begingroup$

It is a problem with the language used, continuity is not determined just by the function but also it requires to specify the domain to be specified.

The function is not defined at $x=2$ so it is meaningless to say it is continuous at 2. As Daniel pointed out, the discontinuity is removable. But that is something that has nothing to do with the original function, the value of function at x=2 has to be explicitly added.

How ever the function is continuous on $(-\infty , 2) \cup (2,+\infty) $ by default or over any interval not containing 2.

Your lecturer was not wrong , but she was incomplete, she should have mentioned the domain of consideration.

Also you have to understand that the two functions $f_2(x)=\frac{x^2-x-2}{x-2}$ and $f_1(x)={x+1}$ are not the same. To see that at x=2 , $f_1(2)=3$ but $f_2(2)$ is undefined, so the two functions are two different things, their graphs are two similar to see that.

PS : Thanks to DamianSobota for correction.

$\endgroup$
2
  • 1
    $\begingroup$ You meant: $f_1(x)=x+1$ and so $f_1(2)=3$. $\endgroup$
    – Damian Sobota
    Nov 28, 2011 at 1:46
  • $\begingroup$ @DamianSobota : Yes, thank you for correction. $\endgroup$
    – jimjim
    Nov 28, 2011 at 2:15
0
$\begingroup$

To talk about contiunity of a function $f$ on a given point $a$ , the function has to be defined on this point and the neighborhood of this point in order to evaluate the limit of $~f~$ when $~x\rightarrow a$ and compare it with $f(a)$ , if the limit is equal to $f(a)$ then it's continuous, otherwise itsn't continuous .

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.