Continuity of this function at a point let $f(x)=x^2$ defined on $[0,1]$
my question may seem silly but i am really confused about it.
if we want to prove the continuity of $f$ at $x=1$ then we should have:
$\lim f(x)$ as x tends to $1_+ = \lim f(x)$ as $x$ tends to $1_- = f(1)$
ok we have $f(1)=1^2=1$ and $\lim f(x)$ as $x$ tends to $1_- = \lim x^2$ as $x$ tends to $1_- =1$
but what about $\lim f(x)$ as $x$ tends to $1_+$ ???? we dont have a definition of $f$ when $x$ tends to $1$ from the right so what shall we put instead of $f(x)$ when we say that $x$ is approaching $1$ from the left??(note that we substituted $f(x)$ by $x^2$ when searching for the $\lim$ of $f$ as $x$ approaches $1$ from the right since in this case $x \in [0,1]$ and therefore $f$ is defined at $x$)
 A: The only thing you have to prove is that when $x \rightarrow 1$ (from the left), then $f(x) \rightarrow f(1)$. That's because $f(x)$ is not defined for $x>1$. 
The definition for continuity you have in mind, and which you want to use, only makes sense for a point which is internal for the interval (or for the set) over which f is defined. So it is not applicable in this case. But, e.g., it's applicable if you want to prove the continuity of $f$ in the point $x_0=1/3$ or $x_0=1/2$. 
There are 2 widely-used definitions which you can use and which make sense in both cases (for internal but also for boundary points $x_0$ for the domain of $f$):
Def.1: the function f said to be continuous at a point $x_0$, if for any sequence $x_1, x_2, ... x_n, ... $ (where $x_k$ are from the domain of $f$) which is convergent and has a limit of $x_0$, the respective sequence $f(x_1), f(x_2), ..., f(x_n), $ ... is also convergent and its limit is $f(x_0)$.
Def.2: the function $f$ is said to be continuous at a point $x_0$, if for any $\epsilon > 0$, one can find a $\delta>0$ such that when $|x-x_0|<\delta$, then $|f(x)-f(x_0)|<\epsilon$.  
The second one is more popular of course. So ... just use one of these two definitions, and you will have no troubles or doubts.
A: I disagree with all the comments you received: there are two possible definitions, I believe:


*

*Either you choose to check continuity approaching the boundary points only from inside (in this case from the left)

*Or you define continuity in a closed set in the following way (this is more common in my experience): $f$ is continuous in $F$ (not necessarily open) if there exist a neighborhood $U$ of $F$ (hence $U$ open, $F \subset U$) and a function $g$ that is continuous on $U$ and satisfies $g_{|_F} = f$

