Consider the functor $C:\mathbf{Top}^{op}\to \mathbf{Ring}$ that sends an object $X$ to a continuous function $X\to \mathbb R$. Consider also the composition $U\circ C$ where $U:\mathbf{Ring}\to\mathbf{Set}$ is a forgetful functor. This composition should be representable, i.e., should be isomorphic to $\mathbf{Top}(-,Y)$ (or $H_Y$) for some topological space $Y$.
To find this $Y$, I expanded what the above means, and the expanded version says that there should be a bijection $$\{\text{functions } X\to \mathbb R \} \leftrightarrow \{\text{continuous functions } X\to Y \}.$$
So it's natural to take $Y$ to be $\mathbb R$. Then the correspondence above is supposed to send a function $f:X\to\mathbb R$ to the same function $f$ that's regarded as continuous map. But why does this map $f$ have to be continous? It looks like this argument implies that any function $X\to \mathbb R$ has to be continuous, which doesn't make sense.
Your functor $C$ sends an object $X$ to the ring of continuous functions $X \to \mathbb R$. Then $U\circ C$ sends $X$ to the set of continuous functions $X\to\mathbb R$. Therefore, the left hand side should be $$ \{\textrm{continuous functions }X\to\mathbb R\}\,. $$