This question comes from one of my exercises, which asks to prove that a mapping on a topological space is continuous if and only if it preserves cluster points. That is to say, for a topological space $X$, $Y$, and a mapping $f:X\to Y$, suppose $\xi:D\to X$ is a net on $X$ and $x$ is a cluster point of this net, then $f(x)$ is a cluster point of the net $f\circ \xi$ on $Y$.
Now that we know a mapping on a topological space is a continuous mapping if and only if for any net $\xi:D\to X$ on $X$, if $\xi\to x$, then the net $f\circ \xi$ on $Y$ converges to $f(x)$, and if $x$ is a cluster point of the net $\xi$ on $X$, then there exists a subnet $\xi\circ\eta$ of $\xi$ such that $\xi\circ\eta\to x$, it is clear that preserving cluster points follows from preserving limits.
But I have no idea how to prove the other direction, and I even began to doubt its correctness because intuitively, the property of preserving continuity is obviously stronger than the property of preserving cluster points. Are they really equivalent? Or have I made a logical mistake in the relationship between these three propositions?
The key idea you seem to be missing is that if a net fails to converge to a point $x$, then there is a subnet such that $x$ is not a cluster point (since there must be some neighborhood $U$ of $x$ which the net is frequently not in so there is a subnet that is never in $U$). So suppose $f:X\to Y$ fails to preserve limits, so there is some net $\xi:D\to X$ which converges to $x\in X$ such that $f\circ\xi$ does not converge to $f(x)$. Then there is a subnet $f\circ\xi\circ\eta$ of $f\circ\xi$ which does not have $f(x)$ as a cluster point. Since $\xi\circ\eta$ still converges to $x$ (and in particular has $x$ as a cluster point), this shows that $f$ fails to preseve clulster points.