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Such functions can be characterized using the notions of effective domain and epigraph. In this way, we study important concepts such as fuzzy indicator function and fuzzy infimal convolution for extended fuzzy-valued functions. Finally, we introduce the concept of directional generalized derivative for extended above functions and its properties. Eventually, we give a practical example that will illustrate well the directional <jats:inline-formula><g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><g:mi>g<\/g:mi><\/g:math><\/jats:inline-formula>-derivative for the extended fuzzy-valued convex function.<\/jats:p>","DOI":"10.1155\/2024\/8335864","type":"journal-article","created":{"date-parts":[[2024,4,10]],"date-time":"2024-04-10T22:05:09Z","timestamp":1712786709000},"page":"1-14","source":"Crossref","is-referenced-by-count":1,"title":["The Investigation of Some Essential Concepts of Extended Fuzzy-Valued Convex Functions and Their Applications"],"prefix":"10.1155","volume":"2024","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6673-3560","authenticated-orcid":true,"given":"T.","family":"Allahviranloo","sequence":"first","affiliation":[{"name":"Software Engineering Department, Faculty of Engineering and Natural Science, Istinye University, Istanbul, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9599-8090","authenticated-orcid":true,"given":"M. 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