For a Banach space $X$ the mapping $$\delta_X\colon[0,2]\to[0,1], \varepsilon\mapsto \inf\{1-|x+y|/2\colon |x|\leq 1, |y|\leq 1, \text{and } |x-y|\geq\varepsilon\} $$ is called the modulus of convexity of $X$. I read at multiple places that $\delta_X$ is continuous on $[0,2)$ but I could not find a proof.
So my question is: how to prove that the modulus of convexity is continuous on $[0,2)$?
Of course, when $\delta_X$ is convex it is easy to see that it is continuous on $(0,2)$ but unfortunately this is not the case in general.
It seems that the original proof of this result is due to V. I. Gurariĭ and published in the Russian paper "V. I. Gurariĭ: Differential properties of the convexity moduli of Banach spaces, Mat. Issled. 2 (1967), no. 1, 141–148".
Unfortunately it seems that this paper is hard to obtain.
The result is also a special case of the results on the modulus of $k$-convexity presented in Lim, Teck-Cheong: On moduli of k-convexity. Abstr. Appl. Anal. 4 (1999), no. 4, 243–247