 
Summary: 18.014ESG Notes 1
Pramod N. Achar
Fall 1999
1 Limits and Continuity
The notation limxp f(x) = L means that for any > 0, you can choose a > 0 such that whenever
0 < x  p < , we are guaranteed that f(x)  L < . N.B.: We do not care how f behaves at x = p, just
near it. (f need not even be defined at p.)
To say that f is continuous at p is to say that limxp f(x) = f(p). Another way of saying this is that for
any > 0, you can choose a > 0 such that whenever x  p < , we are guaranteed that f(x)  f(p) < .
The most important difference between the statements "f has a limit as x approaches p" and "f is continuous
at p" is that in the latter statement, we do care about how f behaves at p.
A function can be either continuous or discontinuous at each point of its domain. At points not in its
domain, the function is neither continuous nor discontinuousjust undefined.
Whenever f : A B has the property that it is onetoone (i.e. if x1 = x2 then f(x1) = f(x2)) and
onto (i.e. for every y0 B there is an x0 A such that f(x0) = y0), we can define the inverse function
f1
: B A of f1
.
In particular, if f : A R is strictly monotonic, then it is onetoone. If B is the image of f, we can
restrict its codomain to B; now f : A B is onetoone and onto, so it has an inverse f1
