Continuous Functions
Read this section for an introduction to what we mean when we say a function is continuous. Work through practice problems 1 and 2.
Which Functions Are Continuous?
Fortunately, the situations which we encounter most often in applications and the functions which model those situations are either continuous everywhere or continuous everywhere except at a few places, so any result which is true of all continuous functions will be true of most of the functions we commonly use.
Theorem: The following functions are continuous everywhere, at every value of :
(a) polynomials, (b) and
and (c)
.
Proof: (a) This follows from the
Substitution Theorem for Polynomials and the definition of continuity.
(b) The graph of (Fig. 6) clearly indicates that
does not have any holes or breaks so
is continuous everywhere. Or we could justify that result analytically:
(recall from section that
and
)
so is continuous at every point. The justification of
is similar.
(c) . When
, then
and its graph (Fig. 7) is a straight line and is continuous since
is a polynomial function. When
, then
and it is also continuous. The only questionable point is the "corner" on the graph when
, but the graph there is only bent, not broken:
A continuous function can have corners but not holes or breaks (jumps).
Several results about limits of functions can be written in terms of continuity of those functions. Even
functions which fail to be continuous at some points are often continuous most places.
Theorem: (a) A rational function is continuous except where the denominator is .
(b) Tangent, cotangent, secant and cosecant are continuous except where they are undefined.
(c) The greatest integer function is continuous except at each integer.