| Sažetak | U prvom poglavlju definirali smo omeđenu funkciju te smo zatim definirali pojmove gornje i donje Darbouxove sume funkcije na segmentu. Zatim smo definirali donji i gornji integral funkcije \(f\) na segmentu \([a,b]\), tj. \(I_*(f;a,b)\) i \(I^*(f;a,b)\) te pojam integrabilne funkcije na segmentu. Dali smo primjere integrabilnih funkcija te dokazali da vrijedi: \(I_*(f;a,b)=I_*(f;a,c)+I_*(f;c,b)\) pri čemu je \(a<b<c.\) Nadalje, dokazujemo da isto vrijedi za gornje integrale: \(I^*(f;a,b)=I^*(f;a,c)+I^*(f;c,b)\) gdje smo koristili jednakost \(I^*(f;a,b)=-I_*(-f;a,b).\) Na kraju smo dokazali da za svaku donju i gornju Darbouxovu sumu vrijedi \(s \leq S\), gdje su \(s\) i \(S\) donja i gornja Darbouxova suma funkcije \(f\), što nas je dovelo do zaključka: \(I_*(f;a,b)\leq I^*(f;a,b)\). U drugom poglavlju proučavali smo neprekidne funkcije. Dokazali smo da je svaka neprekidna funkcija na segmentu uniformno neprekidna. Nadalje, dokazali smo da je svaka neprekidna funkcija na segmentu omeđena te da je, čak štoviše, integrabilna. Dali smo primjer funkcije na segmentu koja je integrabilna, ali nije neprekidna. Na kraju smo dokazali da ako je \(f(x) \leq g(x), \forall x \in [a,b]\), onda vrijedi: \(\int \limits_ a^b f \leq \int \limits_ a^b g\). Dokazali smo i da ako su funkcije \(f\) i \(g\) neprekidne i takve da vrijedi \(f(x_0) <g(x_0)\) za neki \(x_0 \in [a,b]\), onda vrijedi: \(\int \limits_ a^b f < \int \limits_ a^b g\). |
| Sažetak (engleski) | In the first chapter, we defined a bounded function and then introduced the concepts of upper and lower Darboux sums for a function on a segment. We then defined the lower and upper integrals of a function \(f\) on a segment \([a,b]\), denoted by \(I_*(f;a,b)\) and \(I^*(f;a,b)\), as well as the concept of an integrable function on a segment. We provided examples of integrable functions and proved the following statement: \(I_*(f;a,b)=I_*(f;a,c)+I_*(f;c,b)\) where \(a<b<c.\) Furthermore, we showed that the same holds true for the upper integrals: \(I^*(f;a,b)=I^*(f;a,c)+I^*(f;c,b)\), utilizing the equality \(I^*(f;a,b)=-I_*(-f;a,b).\). Finally, we demonstrated that for every lower and upper Darboux sum, \(s \leq S\), where \(s\) and \(S\) represent the lower and upper Darboux sums of the function \(f\) . This led us to the conclusion: \(I_*(f;a,b)\leq I^*(f;a,b)\). In the second chapter, we studied continuous functions. We proved that every continuous function on a segment is uniformly continuous. Furthermore, we established that every continuous function on a segment is bounded and, moreover, integrable. We provided an example of a function on a segment that is integrable but not continuous. Finally, we proved that if \(f(x) \leq g(x), \forall x \in [a,b]\), then \(\int \limits_ a^b f \leq \int \limits_ a^b g\). We have proven that if functions \(f\) and \(g\) are continous and satisfy the condition \(f(x_0) <g(x_0)\) for some \(x_0 \in [a,b]\), then \(\int \limits_ a^b f < \int \limits_ a^b g\). |
