In my book it gave two informal explanation for the concept of continuity. I had doubt in the second explanation but I cleared it by asking it here. The explanation is ,
Suppose a function f has the value f(p) at a certain point p. Then f is said to be continuous at p if at every nearby point x the function value f(x) is close to f(p).
My doubt is that what is range of values that the word 'nearby' supposed to represent because if we keep p=3 both 2 and 4 are near to 3 , 2.5 and 3.5 is also near to 3 and many more numbers are near to 3 I just want to know how near is nearby?
The book name is 'Calculus', Author is Tom Apostol.
It's as near as it needs to be. It should be seen as kind of a challenge. If you prescribe a distance $\varepsilon$ around $f(p)$, there is as distance $\delta$ around $p$ such that if $x$ is at less than $\delta$ from $p$, then $f(x)$ is at less than $\varepsilon$ from $f(p)$.
Actual value for $\delta$ depends on $\varepsilon$, on $p$, and on $f$.