304 Chapter 6 • Differentiable Functions
- Find the values of the constants a and b for which the function
j(x) = { x2
~f x:::;^3 ,} is differentiable at 3.
ax+ b if x > 3- For each of the following functions, answer these questions: Where is f
continuous? Where is f differentiable? What is the formula for f'(x)?
Where is f' continuous? (Explain using graphs; omit proofs.)
(a) f(x) = x + lxl (b) f(x) = xlxl (c) f(x) = I sin xi
(d) f(x) = Ix -ll +Ix+ l l (e) f(x) = x lxJ (f) f(x) = x - lxJ
where lxJ =the greatest integer^2 :::; x.- (a) Prove that the function f(x) = lx^31 is differentiable everywhere. What
is f' (O)?
(b) Prove that the function f(x) = ijX is not differentiable at 0, even
though it is continuous there. - Prove that the function f(x) = { xr sin(~) ~f x # O} is
0 if x=O
(a) continuous from the right at 0 {::} r > O;
(b) differentiable from the right at 0 {::} r > 1.
10. Give alternate definitions of f_(xo) and f~(xo) along the lines of Defini-
t ion 6.1.6.- Prove Theorem 6.1.13.
12. Prove that if f is differentiable from the left (or right) at x 0 , then f is
continuous from the left (or right) at xo.- Find an example of a function f for which f(x 0 ) exists, lim f'(x) and
x--+xQ
lim f'(x) exist and are equal, but f'_(xo) and f~(xo) do not exist.
x--+xci - Show by example that it is possible for both f'_ (x 0 ) and f~ (x 0 ) to exist
(and be equal), even when lim f'(x) and lim f'(x) do not exist.
x->x 0 x->xt
1 5. Use Theorem 6.1.14 to prove that if f is differentiable on some deleted
neighborhood of xo and continuous at x 0 , and lim f'(x) = lim f'(x),
x->x 0 x->xt
then f is differentiable at xo and f' ( x 0 ) = lim f' ( x).
x-+xQ - The "greatest integer function" LxJ is defined in Example 5.2.16.