{"id":{"repo_id":"mo-state","oai_identifier":"oai:bearworks.missouristate.edu:theses-1861"},"canonical_url":"https://search.dev.ndltd.org/etd/mo-state/oai:bearworks.missouristate.edu:theses-1861","repository":{"repo_id":"mo-state","name":"Missouri State University","base_url":"https://bearworks.missouristate.edu/do/oai/"},"display":{"title":"Differentiability, Continuity, and Existence of Limits","abstract":"In this thesis, we first demonstrate in various aspects the existence of everywhere continuous nowhere differentiable functions. We then address the question: could there exist a function defined on R that has a limit at each point of R, but fails to be continuous at any point? We show that the answer to this question is \"No\", and we establish the stronger result that if a function defined on the interval [a,b] has a limit at each point of a dense subset of [a,b], then the set of points where the function is continuous is dense, uncountable, and has the same cardinality as R.","abstract_html":"In this thesis, we first demonstrate in various aspects the existence of everywhere continuous nowhere differentiable functions. We then address the question: could there exist a function defined on R that has a limit at each point of R, but fails to be continuous at any point? We show that the answer to this question is "No", and we establish the stronger result that if a function defined on the interval [a,b] has a limit at each point of a dense subset of [a,b], then the set of points where the function is continuous is dense, uncountable, and has the same cardinality as R.","abstract_has_math":false,"creators":["Millett, Julie Ann"],"institution":null,"degree_name":"Master of Science in Mathematics","degree_level":"Masters","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Xingping Sun"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1994,"date_issued":"1994-12-01T08:00:00Z","date_published":"1994-12-01T08:00:00Z","updated_at":"2026-07-24T03:15:54Z","subjects":["Mathematics"],"languages":[],"rights":["© Julie Ann Millett"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://bearworks.missouristate.edu/theses/860","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Xingping Sun"]},{"key":"dc:creator","label":"Author","values":["Millett, Julie Ann"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© Julie Ann Millett"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://bearworks.missouristate.edu/theses/860"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we first demonstrate in various aspects the existence of everywhere continuous nowhere differentiable functions. We then address the question: could there exist a function defined on R that has a limit at each point of R, but fails to be continuous at any point? We show that the answer to this question is \"No\", and we establish the stronger result that if a function defined on the interval [a,b] has a limit at each point of a dense subset of [a,b], then the set of points where the function is continuous is dense, uncountable, and has the same cardinality as R."]},{"key":"dc:title","label":"Title","values":["Differentiability, Continuity, and Existence of Limits"]}]}],"canonical_facts":{"dc:contributor":["Xingping Sun"],"dc:creator":["Millett, Julie Ann"],"dc:description.abstract":["In this thesis, we first demonstrate in various aspects the existence of everywhere continuous nowhere differentiable functions. We then address the question: could there exist a function defined on R that has a limit at each point of R, but fails to be continuous at any point? We show that the answer to this question is \"No\", and we establish the stronger result that if a function defined on the interval [a,b] has a limit at each point of a dense subset of [a,b], then the set of points where the function is continuous is dense, uncountable, and has the same cardinality as R."],"dc:identifier":["https://bearworks.missouristate.edu/theses/860"],"dc:rights":["© Julie Ann Millett"],"dc:subject":["Mathematics"],"dc:title":["Differentiability, Continuity, and Existence of Limits"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Masters"],"thesis:degree_name":["Master of Science in Mathematics"]},"updated_at":"2026-07-24T03:15:54Z"}