1. ## clarification

i was just wondering which is correct:

a function f is defined on an interval I is contonuous if lim as x->c for f(x) is f(c) for all x in I or for all c in I.

i thought it should be the former but the book that im reading states that its the latter.

thanks

2. It is the latter.

Fernando Revilla

3. thanks for the clarification..but may i know why, if for this case, it is the latter then for this case:
a function f is cont at c if every seq of points x_n in I such that lim as n->infinity for x_n is c if the lim n->inifinity f(x_n) is f(c).
in this case, we are talking about x_n in I and not for all c in I.

i hope im making sense

4. Originally Posted by alexandrabel90
thanks for the clarification..but may i know why, if for this case, it is the latter then for this case:
a function f is cont at c if every seq of points x_n in I such that lim as n->infinity for x_n is c if the lim n->inifinity f(x_n) is f(c).
in this case, we are talking about x_n in I and not for all c in I.

i hope im making sense
Think about what you're saying in a simpler sense. This $\displaystyle x$ is a "dummy" variable, in the same sense as the $\displaystyle x$ in $\displaystyle \displaystyle \int_a^b f(x)\text{ }dx$. What's really fixed is the $\displaystyle c\in I$. Think about your definition you gave in terms of sequences. A function is continuous on $\displaystyle I$ if at every $\displaystyle \mathbf{c}\in I$ that's true. No?