The discussion centers on the continuity of functions, specifically examining the function f(x,y) = x/y, which is continuous everywhere in its domain except where y = 0. It is established that a function can be defined on a domain yet still exhibit points of discontinuity, as illustrated by the piecewise function f(x) = 1 if x = 0 and f(x) = 0 if x ≠ 0, which is not continuous at x = 0 despite being defined for all real numbers. This highlights that continuity is not solely determined by the function's domain.
PREREQUISITESStudents of mathematics, particularly those studying calculus and real analysis, as well as educators seeking to deepen their understanding of function continuity and its exceptions.
No. E.g. take ##f(x) = \begin{cases} 1 & \text{ if } x= 0\\ 0 & \text{ if } x\neq 0 \end{cases}##. It's defined for all real numbers, so its domain is ##\mathbb{R}## whereas it is not continuous at ##x=0##. Of course you can easily find more complicated examples.0kelvin said:Take for ex f(x,y) = x/y. Domain is all (x,y) except for y = 0. It's continuous everywhere except for y = 0. Is this always the case? The function is continuous everywhere in its domain?