My Calculus Professor (Tony Tromba, UC Santa Cruz, Fall 1981) dropped the last example on us at the end of a Friday lecture to give something to snack on during Happy Hour.
One characterization I personally prefer, in the case where x0 is not an isolated point of the domain, is where continuity of f at x0 is defined true if and only if lim f(x) - f(x0) (x -> x0) = 0. This superficially may seem like an unnecessarily complicated way of characterizing continuity, but this is actually a very useful and extremely elegant characterization, because it makes for an intuitive and simple segue into uniform continuity, making the connection between continuity everywhere and uniform continuity almost trivial, and analogous to the connection between pointwise convergence everywhere and uniform convergence. Unexpectedly, it also creates a very nice segue into defining differentiability later on, and other types of continuity, such as Lipschitz continuity.
When we talking about differentiability it is easy to under this is defined on an open interval because in the definition of derivative f(x+h) defined for only interior point but why we use closed interval in tge continuity definition. Same problem occour here also?
Could be possible that in the page 57 is there a mistake where "Then f (xn) = 0 for all n ∈ N and thus limn→∞ xn = 0 != f(x0) = 1" why is the limn→∞ xn = 0 instead of limn→∞ f(xn) = 0? (same question for the second case? Thanks you again!
4:28 you first say that the left limit is different from the right limit. Then, you say that the limit on the point does simply not exist. So, which limits exist now and what is their value? I see it rather that the limits exist but the one is different from the function value at 0.
@@brightsideofmaths Thanks for your quick reply! Do you mean limit from the left (right) when you say approximation from the left (right)? What is then the 'overall limit'?
