To show elements of a set are bounded satisfying certain property.

Let S = { f : $\Bbb R\rightarrow\Bbb R$ | $\exists$$\epsilon$> 0 such that $\forall\delta > 0, |x - y| <\delta\implies |f(x) - f(y) |<\epsilon$ }. Then

(a) S = { f : $\mathbb{R}$$\rightarrow$$\mathbb{R}$ | f is continuous }

(b) S = { f : $\mathbb{R}$$\rightarrow$$\mathbb{R}$ | f is uniformly continuous }

(c) S = { f : $\mathbb{R}$$\rightarrow$$\mathbb{R}$ | f is bounded }

(d) S = { f : $\mathbb{R}$$\rightarrow$$\mathbb{R}$ | f is constant }

If we take Dirichlet function then option a, b, and d will be discarded but how to prove or disprove boundedness of $f$ ?