0

In my book it gave the informal definition of continuity as

If we let $x$ move toward $p$, we want the corresponding function values $f(x)$ to become arbitrarily close to $f(p)$, regardless of the manner in which $x$ approaches $p$.

I don't understand what they mean when they say "regardless of the manner in which $x$ approaches $p$". I need some assistance.

Blue
  • 75,673
Daniel Joseph
  • 59
  • 1
    On $\mathbb R$ it means you can approache either from the left or from the right; in $\mathbb R^2$ you can approache as you want inside a disk of center $p$ and so on according your respective neighborhood. – Piquito May 22 '22 at 11:58
  • When you quote a book, it's good to also give it's title & author. Is this a calc book? Is it discussing multi-variable calculus? Limits in multi-variable calculus need to exist independent of the path taken as the point is approached, which is true in general - but in single variable calc there are only two paths, where as in multi-variable calc there are infinitely many. – Joe May 22 '22 at 12:02
  • @Joe- Sorry, I didn't consider it, the title is "Calculus" , Author- Tom Apostol. I don't know what multi-variable calculus is but judging by the way you say single variable calculus, I am studying single variable calculus – Daniel Joseph May 22 '22 at 13:47
  • single variable calculus predominantly studies functions from $\mathbb{R}$ to $\mathbb{R}$. Multi-variable calculus studies functions from $\mathbb{R}^{n}$ for $n>1$, e.g. $f(x,y)= \sqrt{x^2 + y^2}$. – Joe May 22 '22 at 15:10

1 Answers1

1

I guess what they mean is that regardless of what sequence $\{x_n\}$ such that $\lim_{n\to\infty} x_n=p$ that you choose, you have that $\lim_{n\to \infty} f(x_n)= f(p)$. It just means that if $x$ is close to $p$, then $f(x)$ is close to $f(p)$.

user202542
  • 851