Can anyone guide me for this question?

  • Thread starter Unusualskill
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In summary, the conversation discusses the proof of continuity for a function f(x) which is defined differently for rational and irrational values of x. The proof involves finding a point c at which sin(c) = cos(c) and using the epsilon-delta definition to show that f(x) is continuous at c. The hints provided in the conversation suggest finding a delta value that satisfies the conditions for both sin(x) and cos(x) to show that f(x) is continuous at c.
  • #1
Unusualskill
35
1
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.
 
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  • #2
You were given those "hints", right? So where is your work? Have you found c such that sin(c)= cos(c)?

Why was this posted under "Differential Equations"?
 

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