Problem: Let $X$ and $Y$ be two topological spaces and $f:X \to Y$ be a continuous function. Let $x$ be a limit point of $A$. Then is it necessary that $f(x)$ is a limit point of $f(A)$?
Solution: We recall that $x$ is a limit point of $A$ if every open set $U \subseteq X$ containing $x$ must contain a point of $A$ other than $x$.
Let $X = Y = \mathbb{R} $ and consider the constant function
\[
f: \mathbb{R} \to \mathbb{R} , \quad f(x) = 0, \ \forall\ x \in \mathbb{R} .
\]
Take $A = [0,1]$. Then the limit points of $A$ are $[0,1]$, whereas the limit point of $f(A) = \{ 0 \} $ is empty. Therefore, the $f(x)$ need not be a limit point of $f(A)$.