The purpose of this study was to determine how undergraduates comprehend the theoretical relations between differentiability and integrability concepts within the frame of proof schemes in analysis courses. The study participants were 172 freshmen from three different mathematics departments in Turkey. A questionnaire was used to evaluate the general picture on propositional knowledge about the concepts of continuity, differentiability, and integrability. Afterwards, participants' written answers on the concepts of differentiability and integrability were analysed, particularly in terms of proof schemes usage. The data collected during the study were analysed and interpreted using a classification method and descriptive statistics. Results indicated that many students who answered the propositions correctly could not use valid arguments for their answers. Most of the arguments used by undergraduates in the external proof scheme based on procedural knowledge and described with reference to authority did not have the content to be evaluated as proof. Although analytical proofs were rarely used by undergraduates, more valid arguments were constructed in such proof approaches. The role of definition and propositional knowledge, and also awareness of counterexample, were discussed in order to construct successful theoretical relations in a teaching environment.