Show that if $x$ is a limit point of $A\subset X$ and $f:X \rightarrow Y$ is continuous then $f(x)$ is a limit point of $f(A)$

As the question was answered in comments and there is no one who added a correct answer to close the question. I'm adding this answer (I got the idea of the answer from This comment)

The statement is false, here is a concrete example:

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ where, $f(x)=2$ for all $x\in \mathbb{R}$. So $f$ is a constant function.

Note that we denote the set of limit points of $H$ by $H'$.

Let $A = [0,1]$. So, $A' = [0,1]$.

$f(A)=f([0,1])=\{2\}$, but $f(A)'=\emptyset$ because {$2$} has no limit points as if $x\not = 2$ is a real number then there is always an interval $C$ containing $x$ but not contaning $2$.

So Although $1$ is a limit point of $A$, $f(1)=2$ is not a limit point of $f(A)=\{2\}$.

As the question was answered in comments and there is no one who added a correct answer to close the question. I'm adding this answer (I got the idea of the answer from comments).