3. Dene the function
ha(x) =
(
xa if x > 0;
0 if x 0:
where a is a real constant.
(a) (3 points) For what values of a is ha continuous at 0?
(b) (4 points) For what values of a is ha dierentiable at 0?
(c) (3 points) For what values of a is the derivative function h0
a continuous on R?
Make sure to show all your work.
4. Let K R be compact and f : K ! R be continuous. In the following problem, we will
show that for every ” > 0, there exists M(“) > 0 such that for all x; y 2 K we have
jf(x) .. f(y)j M(“)jx .. yj + “:
(a) (3 points) We will argue by contradiction. Show that if the above statement does
not hold, there exists “0 > 0 and sequences (xn); (yn) in K such that
jf(xn) .. f(yn)j > njxn .. ynj + “0:
Hint: Proceed inductively.
(b) (2 points) Given your sequences (xn) and (yn) as above, show that there are sub-
sequences (xnk) and (ynk ), such that xnk ! x0 2 K and ynk ! y0 2 K.
(c) (5 points) Using parts (a) and (b), arrive at a contradiction by considering the
cases when x0 = y0 or x0 6= y0.
6. (9 points) Prove using the ” denition of continuity that f(x) = x3 3×2+x 1
is continuous on R.
Hint: You may nd the following inequality from helpful: for any a; b 2 R,
we have ab 1
2a2 + 1
2b2.
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