A constant function is continuous and breaks both image and preimage cardinality claims at once

1. A.The set of all finite subsets of $\mathbb{Z}$ is uncountable.
2. B.Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function. If $S \subseteq \mathbb{R}$ is a countably infinite set, then f(S) is a countably infinite set.✓ Correct
3. C.Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Then there exists an uncountable subset $T \subseteq \mathbb{R}$ such that $f(T) \subseteq \mathbb{R}$ is uncountable.
4. D.Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. If R ⊆ image(f) is a countable set, then f⁻$^{1}(R)$ is countable.

The trap it tests

There is a canonical object that settles this. Recognising it is the whole question.