#1.1 Suppose that
(H1) f is continuous on [0,1]
(H2) g is continuous on [1,2]
(H3) f(1) = g(1).
(H4) h(x) = f(x) when 0 < x < 1 and h(x) = g(x) when 1 < x < 2.
(a) Give an epsilon-delta proof that h is continuous at 1.01.
(b) Give an epilon-delta proof that h is continuous at 1.
#1.2 Suppose that
(H1) f is strictly monotone increasing on [0,2]
(H2) A = {f(x) : 0 < x < 2}.
(a) Explain carefully why A must have a greatest lower bound.
(b) Show that lim(x approaches 0) f(x) must exist.
(c) Show that f(0) < lim(x approaches 0) f(x) .
#1.3 Suppose that
(H1) f is de ned on (-2,2) and
(H2) f is di erentiable at 0 and f'(x) = 3:
Do not assume that f is diff erentiable anywhere else.
Show that there must exist a positive delta such that
f(x) > f(0) + 2.9 (x - 0) whenever 0 < x < delta .
(H1) f is continuous on [0,1]
(H2) g is continuous on [1,2]
(H3) f(1) = g(1).
(H4) h(x) = f(x) when 0 < x < 1 and h(x) = g(x) when 1 < x < 2.
(a) Give an epsilon-delta proof that h is continuous at 1.01.
(b) Give an epilon-delta proof that h is continuous at 1.
#1.2 Suppose that
(H1) f is strictly monotone increasing on [0,2]
(H2) A = {f(x) : 0 < x < 2}.
(a) Explain carefully why A must have a greatest lower bound.
(b) Show that lim(x approaches 0) f(x) must exist.
(c) Show that f(0) < lim(x approaches 0) f(x) .
#1.3 Suppose that
(H1) f is de ned on (-2,2) and
(H2) f is di erentiable at 0 and f'(x) = 3:
Do not assume that f is diff erentiable anywhere else.
Show that there must exist a positive delta such that
f(x) > f(0) + 2.9 (x - 0) whenever 0 < x < delta .
