- #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.
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.