Summary: 18.014ESG Notes 1
Pramod N. Achar
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 discontinuous--just undefined.
Whenever f : A B has the property that it is one-to-one (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
: B A of f1
In particular, if f : A R is strictly monotonic, then it is one-to-one. If B is the image of f, we can
restrict its codomain to B; now f : A B is one-to-one and onto, so it has an inverse f-1