 Continuous real maps from the initial-segment topology

Give $\mathbb{N}$ the topology whose nonempty open sets are $\mathbb{N}$ and the finite initial segments. Every nonempty open set contains $1$, so any two nonempty open sets intersect. A continuous map from this topological space into a Hausdorff space is constant: distinct image points would have disjoint open neighbourhoods, whose inverse images would be disjoint nonempty open sets. This includes continuous real-valued maps.