First Semester in Numerical Analysis with Julia, 2020a

Chapter 1
Introduction
1.1 Review of Calculus
There are several concepts and facts from Calculus that we need in Numerical Analysis. In
this section we will list some definitions and theorems that will be needed later. For the
most part functions in this book refer to real valued functions defined on real numbers R,
or an interval (a, b) ⊂ R.
Definition 1. 1. A function f has the limit L at x 0 , written as lim x→x0 f(x) =L, if for
any ɛ>0, there exists δ>0 such that |f(x) − L| 0 such that |x n − x| N.
Theorem 2. The following are equivalent for a real valued function f :
1. f is continuous at x 0
2. If {x n } ∞ n=1 is any sequence converging to x 0 , then lim n→∞ f(x n )=f(x 0 ).
Definition 3. We say f(x) is differentiable at x 0 if
exists.
f ′ (x 0 ) = lim
x→x0
f(x) − f(x 0 )
x − x 0
6
= lim
h→0
f(x 0 + h) − f(x 0 )
h