Geodesic
Analytic functions are $\Cal I$-density continuous
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 4, pp. 645-652.

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A real function is $\Cal I$-density continuous if it is continuous with the $\Cal I$-density topology on both the domain and the range. If $f$ is analytic, then $f$ is $\Cal I$-density continuous. There exists a function which is both $C^\infty $ and convex which is not $\Cal I$-density continuous.
Classification : 26A21, 26E05, 26E10
Keywords: analytic function; $\Cal I$-density continuous; $\Cal I$-density topology 
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     title = {Analytic functions are $\Cal I$-density continuous},
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Ciesielski, Krzysztof; Larson, Lee. Analytic functions are $\Cal I$-density continuous. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 4, pp. 645-652. http://geodesic.mathdoc.fr/item/CMUC_1994__35_4_a5/