Page 78 - Calculus Demystified
P. 78
Foundations of Calculus
CHAPTER 2
Fig. 2.5 65
SOLUTION
If x< 3 then the function is plainly continuous. The function is undefined
at x = 3 so we may not even speak of continuity at x = 3. The function is also
obviously continuous for 3 <x < 4. At x = 4 the limit of g does not exist—it
is 1 from the left and 11 from the right. So the function is not continuous (we
sometimes say that it is discontinuous)at x = 4. By inspection, the function is
continuous for x> 4.
You Try It: Discuss continuity of the function
x − x 2 if x< −2
g(x) = 10 if x =−2
−5x if x> −2
We note that Theorem 2.1 guarantees that the collection of continuous functions
is closed under addition, subtraction, multiplication, division (as long as we do not
divide by 0), and scalar multiplication.
Math Note:If f ◦ g makes sense, if lim x→c g(x) = , and if lim s→ f(s) = m,
then it does not necessarily follow that lim x→c f ◦g(x) = m. [We invite the reader
to find an example.] One must assume, in addition, that f is continuous at . This
point will come up from time to time in our later studies.
In the next section we will learn the concept of the derivative. It will turn out
that a function that possesses the derivative is also continuous.
