Continuity and Differentiability

from the University of Tokyo 1S1 Mathematical Science Foundation: Differential and Integral Calculus document Continuity and Differentiability

• When a function remains continuous after being differentiated n times, it is said to be n times continuously differentiable.
  • The terminology can be confusing, as being able to differentiate a function continuously n times does not necessarily make it “n times continuously differentiable”.
    • If a function can only be differentiated continuously, it is simply “n times differentiable”.
    • It becomes “n times continuously differentiable” if it remains a continuous function after that.
• $C^n$ function: a function that is n times continuously differentiable.
  • A function that can be differentiated any number of times is called a $C^{\infty }$ function.
  • In high school, we mostly dealt with $C^{\infty }$ functions, but it is possible to construct functions that are only differentiable a finite number of times if done carefully.