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This question already has answers here: Does every continuous map from $\mathbb{Q}$ to $\mathbb{Q}$ extends continuously as a map from $\mathbb{R}$ to $\mathbb{R}$? (3 answers) Closed last year .

In a review of H.R. Pitt's Integration, Measure and Probability, Sir John Kingman wrote,

The author is often careless about details, asserting for instance (on page 105) that a function continuous on the rationals has a continuous extension to the reals.

Using the properties of Cauchy sequences and completeness of $\mathbb R$, I can prove that if $f:\mathbb Q\to\mathbb R$ is uniformly continuous, then there exists a continuous function $g:\mathbb R\to\mathbb R$ such that $f=g$ on $\mathbb Q$.

It follows that any inextensible $f$ cannot be uniformly continuous on $\mathbb Q$. However, I am unable to come up with a concrete example.