Ret Easy 3

Link

• #define continuity at a point
• \[ \lim_{x\to a}f(x) = f(a) \]
  • To ensure that it is defined, connected on both sides, and doesn't have a random point
  • To check for continuity, just check for \(f(a)\), \(\lim_{x\to a}f(x)\), and that they are equal
• Rational functions
  • Are continuous on their domains
    • Basically anywhere they are defined
• Discontinuity types
  • Removable discontinuities
    • Hole in the graph
  • infinite is continuity
    • asymtote
  • jump discontinuity
• Continuity from the right and left
  • Same as definition of continuous, but replace the limit with right and left hand limits respectively

Link - Basically the same thing - Properties of continuous functions (group like bits) - > Let 𝑓 and 𝑔 be continuous functions on an interval 𝐼 , let 𝑐 be a real number and let 𝑛 be a positive integer. The following functions are continuous on 𝐼 . > - Sums/Differences : 𝑓±𝑔 > - Constant Multiples : 𝑐⋅𝑓 > - Products : 𝑓⋅𝑔 > - Quotients : 𝑓/𝑔 (as long as 𝑔≠0 on 𝐼 ) > - Powers : 𝑓𝑛 > - Roots : \(f(x) = \sqrt[n]{x}\) (if 𝑛 is even then 𝑓≥0 on 𝐼 ; if 𝑛 is odd, then true for all values of 𝑓 on 𝐼 .) > - Compositions : Adjust the definitions of 𝑓 and 𝑔 to: Let 𝑓 be continuous on 𝐼, where the range of 𝑓 on 𝐼 is 𝐽 , and let 𝑔 be continuous on 𝐽. Then 𝑔∘𝑓, i.e., 𝑔(𝑓(𝑥)), is continuous on 𝐼. -