{"paper":{"title":"Continuous horizontally rigid functions of two variables are affine","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"M\\'arton Elekes, Rich\\'ard Balka","submitted_at":"2011-09-22T19:57:38Z","abstract_excerpt":"Cain, Clark and Rose defined a function $f\\colon \\RR^n \\to \\RR$ to be \\emph{vertically rigid} if $\\graph(cf)$ is isometric to $\\graph (f)$ for every $c \\neq 0$. It is \\emph{horizontally rigid} if $\\graph(f(c \\vec{x}))$ is isometric to $\\graph (f)$ for every $c \\neq 0$ (see \\cite{CCR}).\n  In an earlier paper the authors of the present paper settled Jankovi\\'c's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \\in \\RR$). Later they proved that a continuous function of two variables is vertically ri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4933","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}