# VC Dimensions in Machine Learning

Hello I'm learning about VC dimensions in machine learning. The class of classifiers $$H$$ where $$h \in H$$ if $$h \in \mathbb{R} \rightarrow \{0,1\}$$ is what I believe is simply binary classifiers (with mapping from the real numbers to 0 or 1). And with $$\mathcal{X} = \mathbb{R}$$, is it true that the VC dimension is $$\infty$$.

My intuition to this is that $$H$$ is a rich set of classifiers that contains any sort of mapping between real numbers and 0 or 1. So for instance both $$(9,0)$$ and $$(9,1)$$ are contained in $$H$$, and so does $$(-3.1,0)$$ and $$(-3.1,1)$$. This means that whichever set of points I pick in the real number space (and no matter how many they are), I could get an adversary to label them with 0s or 1s whichever way they want, and I can still find a classifier in $$H$$ that happens to give the correct labelling for all configurations - since I happen to have the labellings for both 1 and 0 in my $$H$$. So thus, since $$H$$ can shatter an arbitrarily large number of points then I can say $$VC_{dim}(H) = \infty$$?

Am I thinking about it correctly? But something also tells me that I'm wrong because $$(9,0)$$ and $$(9,1)$$ shouldn't both be in the $$H$$ as it maps any real number $$x$$ to two items: 0 and 1, which is contradictory to the definition of a function. So now, I'm debating whether I interpreted the class of classifiers $$H$$ correctly.