Surely in a basic Calculus course it is very common studying limits of functions but surprisingly (at least this is my experience) in Topology it is not common to do this: indeed, between many famous text (Engelking, Nagata, Kelley, Munkres, etc...) only the Bourbaki text sketches limits of functions theory: so when I started to study Topology, I searched many references about limits of functions and in particular on this site I found here an answer where the user TheSilverDoe defines the limit of a function into a topological space as to follow.
Definition - 0
If $f$ is a function from a topological space $X$ to a topological space $Y$ then $y_0\in Y$ is the limit of $f$ as $x\in X$ approaches to $x_0\in X$ if for all neighborhood $V_0$ of $y_0$ there exists a neighborhood $U_0$ of $x_0$ whose image $f[U_0]$ under $f$ is contained into $V_0$.
So with respect this definition the function $f$ is continuous at $x_0$ if and only if $f(x_0)$ is its limit as $x$ approaches to $x_0$ but it seem that this limit is not generally unique. Anyway the Bourbaki text$^0$ gives the following definition
Definition - 1
Let $f$ be a mapping of a set $X$ into a topological space $Y$ and $\mathcal F$ be a filter on $X$. A point $y\in Y$ is said to be a limit point of $f$ with respect to the filter $\mathcal F$ if $y$ is a limit point of the filter base $f(\mathcal F)$
so that then it says that$^1$
"Let $X$ and $Y$ be two topological spaces, $f$ a mapping of $X$ into $Y$, $\mathcal U(x_0)$ the neighborhoods filter in $X$ of a point $x_0\in X$. Instead of saying that $y_0\in Y$ is a limit of $f$ with respect to the filter $\mathcal U(x_0)$ writing $$ y=\lim_{U\in \mathcal U(x_0)} f $$ we use the special notation $$ y=\lim_{i\in I}f(x) $$ saying that $y_0$ is a limit of $f$ at the point $x_0$, or that $f(x)$ tends to $y_0$ as $x$ tends to $x_0$."
Therfore we let to prove that the definitions $0$ and $1$ are equivalent.
So if for any $V_0\in\mathcal V(y_0)$ there exists $U_0\in\mathcal U(x_0)$ such that $$ f[U_0]\subseteq V_0 $$ then $V_0$ lies into the filter $\mathcal F\big(f,\mathcal U(x_0)\big)$ generated by $f$ and $\mathcal U(x_0)$ because the collection $$ f\big(\mathcal U(x_0)\big):=\big\{f[U_0]:U_0\in\mathcal U(x_0)\big\} $$ is a base for this filter. Conversely if $\mathcal F\big(f,\mathcal U(x_0)\big)$ converges to $y_0$ then there exists $f[U_0]\in f\big(\mathcal U(x_0)\big)$ such that $$ f[U_0]\subseteq V_0 $$ because we restate that $f[U_0]$ is a base for $\mathcal F\big(f,\mathcal U(x_0)\big)$.
Now we let to prove this two results which are a generalization of well known propositions about relation between filters or nets with continuity: moreover the arguments used to prove the following propositions are the same used to prove the mentioned propositions.
Proposition 2
A function $f$ from a space $X$ to a space $Y$ converges to $y_0$ as $x$ approaches to $x_0$ if and only if the immage $$ f(\mathcal F_0):=\big\{f[F_0]:F_0\in\mathcal F_0\big\} $$ of any filter $\mathcal F_0$ converging to $x_0$ is a filter base converging to $y_0$.
Proof. So if $y_0$ is the limit of $f$ as $x$ approaches to $x_0$ then for any $V_0\in\mathcal V(y_0)$ there exists $U_0\in\mathcal U(x_0)$ such that $$ f[U_0]\subseteq V_0 $$ but $U_0\in\mathcal F_0$ so that $V_0\in\mathcal F(f,\mathcal F_0)$ and thus $\mathcal F(f,\mathcal F_0)$ converges to $y_0$, where $F(f,\mathcal F)$ is the filter generated by $f(\mathcal F_0)$.
Conversely if $\mathcal F(f,\mathcal F_0)$ converges to $y_0$ for any $\mathcal F_0$ then in particular for $\mathcal F\big(f,\mathcal U(x_0)\big)$ converges to $y_0$ and thus by what above observed we conclude that $y_0$ is the limit of $f$ as $x$ approaches to $x_0$.
Proposition 3
A function $f$ from a space $X$ to a space $Y$ converges to $y_0$ as $x$ approaches to $x_0$ if and only if the net immage $\big(f(x_\lambda)\big)_{\lambda\in\Lambda}$ of any net $(x_\lambda)_{\lambda\in\Lambda}$ converges to $y_0$ provided that $(x_\lambda)_{\lambda\in\lambda}$ converges to $x_0$.
Proof. So if $y_0$ is the limit of $f$ as $x$ approaches to $x_0$ then for any $V_0\in\mathcal V(y_0)$ there exsits $U_0\in\mathcal U(x_0)$ such that $$ f[U_0]\subseteq V_0 $$ but $(x_\lambda)_{\lambda\in\Lambda}$ converges to $x_0$ so that there exists $\lambda_0\in\Lambda$ such that $$ f(x_\lambda)\in f[U_0]\subseteq V_0 $$ which means that $\big(f(x_\lambda)\big)_{\lambda\in\Lambda}$ converges to $y_0$.
Conversely we suppose that $\big(f(x_\lambda)\big)_{\lambda\in\Lambda}$ converges to $y_0$ for any net $(x_\lambda)_{\lambda\in\Lambda}$ converging to $y_0$ is not the limit of $f$ as $x$ approaches to $x_0$. So in this case there would be exists $V_0\in\mathcal V(y_0)$ such that $$ f[U_0]\nsubseteq V_0 $$ for any $U_0$, that is $$ U_0\cap\big(X\setminus f^{-1}[V_0]\big)\neq\emptyset $$ so that if $\sigma$ is a choice function from $\mathcal P(X)\setminus\{\emptyset\}$ to $X$ then we put $$ x_{U_0}:=\sigma\Big(U_0\cap\big(X\setminus f^{-1}[V_0]\big)\Big) $$ for each $U_0\in\mathcal U(x_0)$. Now it is not hard to verify that the position $$ U_0'\preceq U_0''\longleftrightarrow U_0''\subseteq U_0' $$ for $U_0',U_0''\in\mathcal U(x_0)$ nests $\mathcal U(x_0)$ so that $(x_{U_0})_{U_0\in\mathcal U(x_0)}$ is a net which obviously converges to $x_0$ but $\big(f(x_{U_0})\big)_{U_0\in\mathcal U(x_0)}$ does not converges to $y_0$ because $f(x_{U_0})$ does not lie in $V_0$ for any $U_0\in\mathcal U(x_0)$.
Now we know that $f$ is continuous at $x_0$ if and only if $f(x_0)$ is just the limit of $f$ as $x$ approaches to $x_0$ so that by the last two proposition we infer that $f$ is continuous at $x_0$ if and only if the immage $f(\mathcal F_0)$ of any filter $\mathcal F_0$ convergin to $x_0$ is a filter base converging to $f(x_0)$ and if and only if the net immage $\big(f(x_\lambda)\big)_{\lambda\in\Lambda}$ of any net $(x_\lambda)_{\lambda\in\Lambda}$ converges to $f(x_0)$ provided that $(x_\lambda)_{\lambda\in\lambda}$ converges to $x_0$. Moreover a similar result holds for composition limits.
So theese result show apparently that TheSilverDoe definition works perfectly but instead it seem that this defintion of limit if different form this given by the user HennoBrandsma because the definition we use does not require or imply that any function has only one limit at any point whereas Henno's definition requires just this: so I would like to understand if the TheSilverDoes's definition agrees effectively with Bourbaki's definition and thus I would like to know if the proposition $2$ and $3$ are true and thus if they are well proved because I did not find it anywhere. Finally why only the Bourbaki text I found some results about limit: perhaps limit are not studied in Topology becasue any result can about them can be proved with very similar arguments used for continuity?
So could some one help me, please?
$^0$ See Elements of Mathematics - General topology pt. 1 Chpt. 1 §$7.3$
$^1$ See Elements of Mathematics - General topology pt. 1 Chpt. 1 $§7.4$
