Define functions f and g on [-1,1] by
f(x) = xcos(1/x) if x≠0 and 0 if x = 0
g(x)= cos(1/x) if x≠0 and 0 if x = 0
(These are piecewise defined. I don't know how to type them in here.)
Prove that f is continuous at 0 and that g is not continuous at 0. Explain why these functions are continuous at every other point in [-1,1].
The Attempt at a Solution
Using the definition of continuity,
Suppose g is continuous at x=0.
Since g is continuous for all x in [-1,1], then for all ε>0, there exists a δ > 0 such that |g(x)-g(c)| < ε for all x in [-1,1] that satisfy |x-c| < δ
Let c = 0
Then |g(x)-g(c)| = |cos(1/x)| and since -1 ≤ cos(1/x) ≤ 1 for all x in [-1,1], then |cos(1/x)|≤|2x|<2δ=2/2=1
Then if x=3/[itex]\pi[/itex], |cos(1/x)| = |1/2| < 1/2 which 1/2 cannot be less than itself. So there exists a ε where the continuity definition fails at c=0 and thus g(x) is not continuous on [-1,1].
Is this correct for showing g(x) is not continuous at x=0?
I am unsure how to show f(x) is continuous at x=0 since it is a product of two functions where one of them is not continuous at 0.
I have a theorem that will help me with showing that trig functions are continuous if they are defined on their domain.