[8] 9. Either prove the statement is true or show that it is false.
(a) sin (sin(2)) = 27.
Solution: It is FALSE. The range of sin-' r is (-1,1], hence
sin (sin(27)) = sin (0) = 0
(b) If f is continuous at a, then f is continuous at a.
Solution: It is TRUE. Let g(x) = \f()]. If lim f(x) = f(a), then we have
since the absolute value function is continuous
lim g(x) = lim f() = lim f(x)
= f(a) = g(a)
So, g = \f is continuous at a.
(c) If [g] is continuous at a, then g is continuous at a.
-1 if r <0
Solution: It is FALSE. Let g(x) = 3
· Then, g = 1, so g is continuous at 0
i if >0
by the Continuity Theorems, but
lim g(t) = -1 + lim g(1)
240-
1-
0+
So, g(x) is not continuous at a.
(d) If f is continuous at a, then f is differentiable at a.
Solution: It is FALSE. We know that f(1) = 2 is continuous at 0, but is not differentiable
at 0.
