Definition of Continuity at a Point - Page 3

Questions
  • For each point of discontinuity, determine whether f has a limit. If it does, find it. If it doesn’t, explain why?
  • Show that if #T# is continuous and #A sube Y# is any open set in #Y#, that #T^(-1)(A)# is open?
  • What is the difference between an open interval and a closed interval?
  • #f(x) = 3x^2+2x+1# Does #f(x)# have points of discontinuity over its domain ?
  • Let y=f(x) be a continuous function defined on the closed interval [0,b] with the property that 0<f(x)<b for all x in [0,b]. Show that there exists a point c in (0,b) with the property that f(c)=c. How would set this up?
  • Let #f:[a,b]->RR^+uu{0}# where #int_0^bf(x)dx=0# then #f(x)=0# on #[a,b]# this sentence true or false ?
  • For a given point on a function, does differentiability imply continuity?