Can anyone guide me for this? thank you in advance

1. Sep 20, 2014

Unusualskill

Let f be the function:
f(x) =
sin(x) ; x is element of Q
cos(x) ; x is not element of Q
Prove, using epsilon-delta definition, that there is a point c,which is element of R at which f is continuous.
Hint: Consider c such that sin(c) = cos(c); why does such a c exist? Then,
since you know that sin(x) and cos(x) are continuous at c, for epsilon> 0, you get delta1 >
0 that gives lsin(x)-sin(c)l <epsilon , and also delta2 > 0 that gives lcos(x)-cos(c)l <epsilon .
Now, why does delta = min(delta1; delta2) work to show lf(x)- f(c)l < epsilon.

2. Sep 20, 2014

mathman

Since at each x one of the deltas work then the smaller works for all.

3. Sep 21, 2014

Unusualskill

But how to find that c such that sin(c)=cos(c)?

4. Sep 21, 2014

pasmith

Hint: sin(c) = tan(c)cos(c), and tan is surjective.

Alternatively, consider that the graphs of sin and cos intersect.

In the further alternative (not really necessary since either of the above two are sufficient to conclude that such a $c$ exists and you don't actually need to find an exact value for it), consider the triangle formed by cutting a unit square along a diagonal.

Really, between the hint provided in the question and mathman's response you now have a complete answer to the analysis part of the question. The incidental trigonometry should have been obvious; you should be thoroughly familiar with the basic properties (geometric definitions, graphs, special values) of the trigonometric functions before studying calculus or real analysis.

Last edited: Sep 21, 2014
5. Sep 21, 2014

HallsofIvy

Staff Emeritus
Equivalently, sin(c)= cos(c) is the same as saying tan(c)= 1. You should be able to solve that!

6. Sep 21, 2014

Unusualskill

Erm, what i don understand is that why i have to find such c? n why sin x and cos x will be continuous at that c?eg 45 degree?

7. Sep 21, 2014

Fredrik

Staff Emeritus
You will understand that when you do the final step of the proof that was sketched in the problem statement.