What's "stetig fortsetzbar" in English?

If, say, a function $$f\colon(a,b)\to\mathbb R$$ is continuous and there's a continuous function $$g\colon[a,b)\to\mathbb R$$ such that $$f$$ and $$g$$ agree on $$(a,b)$$, then some German textbooks call $$f$$ "stetig fortsetzbar" in $$a$$, and $$g$$ is called the "stetige Fortsetzung". Of course, "stetig" is for continuous, but is there a standard English term for this concept? Is it maybe "continuously continuable"?

[FWIW, the more general definition is that a function $$f$$ defined on a set $$D$$ is "stetig fortsetzbar" in $$x_0\neq D$$ if $$\lim_{x\to x_0}f(x)$$ (is defined and) exists.]

• 'Continuous extension', I guess. Sep 17, 2022 at 8:13