In "Neighbor Rosicky" by Willa Cather, Doctor Ed Burleigh is the first character mentioned; he is also the last. That is, of course, a deliberate choice by Cather. She uses the young doctor both to introduce us to the protagonist, Anton Rosicky, and to help us bid Rosicky farewell at the end of the story.
In the exchange the two men have in the doctor's office, we meet Rosicky and discover a lot about his temperament. Though he has just been given some serious, even devastating news, Rosicky is not panicked or flustered. Neither is he angry or upset. He is calm and even humorous, characteristics we learn later are essential parts of who he is in all circumstances. There is no panic when he hears Burleigh's pronouncement; instead he smiles and says:
"Well, I guess you ain't got no pills fur a bad heart, Doctor Ed. I guess the only thing is fur me to git me a new one."
Rosicky is gracious and humble, but in his exchange with the doctor we also learn that he is hard-working and stubborn (note the list of things he asks the doctor if he will still be able to do). In fact, everything we later learn about Rosicky as he interacts with his wife, sons, and daughter-in-law as well as in the flashbacks to his time in London are foreshadowed in his meeting with Doctor Burleigh. Their meeting serves as the exposition to the story as well as the inciting action--Rosicky's heart is wearing out and he will die if he overexerts himself.
At the end of the story, after Rosicky has been dead for several weeks, Doctor Burleigh finally takes the opportunity to pay his last respects to the older man. This also serves as the readers' only opportunity to pay their own respects to the kind and gentle man. Without him, we would have no closure.
One other element that Doctor Burleigh's perspective adds to this story is the respect and kind of adoration that he has for the much older man, probably indicative of how others feels about Rosicky, as well. The doctor is always happy to spend some time with the Rosickys, and Burleigh shares his thoughts about why they are happy. At first he wonders why this family "didn't get much ahead." He knows they are hard-working and make a comfortable enough living, and yet they are not much better off than just being out of debt. His conclusion is the conclusion we are also supposed to come to about the Rosickys:
people as generous and warm-hearted and affectionate as the Rosickys never got ahead much; maybe you couldn't enjoy your life and put it into the bank, too.
Although the only thing we really need from Doctor Burleigh in this story is his pronouncement that Rosicky's heart is weak, we get much more than that. We get an honest introduction, characterization, and farewell for a lovely man named Anton Rosicky.