## Transcribed Text

PROBLEM # 1
Let f : [a,
[c, d] be a continuous function. Suppose that g : [a, b] R is another continuous function
such
that
there
exists x1, x2 € [a, b] with the property g(x1) = c and g (x2) = d. Prove that there exists some
xo € [a, b] with f(zo)=g(zo). =
PROBLEM # 2
Let f : [a, b] a, b] be a continuous function with If(x) - f(y) < x - y for all x # y € [a,b]. Prove that
there
exists
a
unique
xo
€
[a,
b]
with
f
(xo)
=
xo,
and
moreover
that
the
iterated
composition
fn
=
fofo..
f
(n times) is such that xn = fn '(x) xo for all x € a, b].
PROBLEM # 3
Consider the following unctionf:R-R:
1
if
x=0;
f(x) =
n 1/2 , if x = m n where m E Z , n € N and gcd(n,m) = 1;
0 , if
Show that f is continuous at all x € R)Q, and discontinuous at all x € Q.
PROBLEM # 4
Let p(x) : R R be a monic odd degree polynomial, i.e. p(x) = Er-sakach where d is odd and ad = 1.
Suppose there exists X1 < x2 with p(x2) <0Show that the polynomial p(x) = x3 + 4x² + x - 1 factors with all real roots.
PROBLEM #5
Let f : D R, and let xo € D. We say "f is left continuous at x0" if whenever In € D and Xn < xo,
then xn
xo implies f (xn) f (xx). Likewise we say "f is right continuous at xo" if whenever xn € D and
xn > xo, then xn
xo implies f (xn)
f(xo).
Show that a function f : D
R is continuous at xo € D iff it is both right and left continuous as xo.
PROBLEM #6
A function f : D R is called lower semicontinuous at xo € D if f < lim inf f(xn) for all sequences
xn € D with Xn
xo € D.
A function f : D R is called upper semicontinuous at xo € D if lim sup f(xn) < f(xo) for all sequences
xn € D with xn
xo € D.
a) Prove that a function f : D
R is continuous at xo € D iff it is both upper and lower semicontinuous
at xo.
b) Prove that a monotone increasing left continuous function (see Problem #5 above) is always lower semi-
continuous.
c) Prove that a monotone increasing right continuous function (see Problem #5 above) is always upper
semicontinuous.
d)
Prove that the function from Problem #3 above is upper semicontinuous at all points.

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