- #1
jgens
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Homework Statement
Prove that the sine function is continuous on its domain.
Homework Equations
N/A
The Attempt at a Solution
I think that I've gotten this right but I would appreciate it if someone checked my solution . . .
Let [tex]\epsilon > 0[/tex]. We define [tex]\delta[/tex] such that,
[tex]0 < |x - a| < \delta = \epsilon[/tex]
Now, by the mean-value theorem of differential calculus, we have that,
[tex]1 \geq \frac{sin(x) - sin(a)}{x - a} \geq -1[/tex]
or similarly,
[tex]1 \geq \left |\frac{sin(x) - sin(a)}{x - a} \right |[/tex]
To complete the proof, we utilize this inequality such that
[tex]0 < |sin(x) - sin(a)| \leq |x-a| < \epsilon[/tex]
As desired. This proves that the sine function is continuous on its domain.