I misposted the equation it should read
I have a question about the proof of this elementary DE, the calculations are easy
-not sure if you can add the integral into the absolute value
is a continuous function of time
-I'm not sure how this is assumed to be continuous in time?
If the absolute value of a function g(t) is constant then g must be constant,
Proof:
If g isn't constant there exists t1,t2 for which g(t1)=c, g(t2)=-c. By the intermediate value theorem g must achieve all values between -c and +c which is impossible if |g(t)|=c
-but if g(t)=x or d then why does it necessarily equal c and -c
You are given that dy/dt exists, y is differentiable so continuous. Any integral is continuous, the exponential of a continuous function is continuous, and the product of two continuous functions is continuous. That's why is continuous.-I'm not sure how this is assumed to be continuous in time?
No, that's not true. The absolute value of "g(t)= 1 if t is rational, g(t)= -1 if t is irrational" is the constant 1.If the absolute value of a function g(t) is constant then g must be constant
Again, no. The function g(x)= 1 if x is rational, 2 if x is irrational has NO pair t1 and t2 such that g(t1)= c, g(t2)= -c.If g isn't constant there exists t1,t2 for which g(t1)=c, g(t2)=-c.
Are you assuming that g is continuous? You didn't say that.By the intermediate value theorem g must achieve all values between -c and +c which is impossible if |g(t)|=c
-but if g(t)=x or d then why does it necessarily equal c and -c
It is true that if the absolute value of a continuous function is constant, then the function is constant.
(But "If g isn't constant there exists t1,t2 for which g(t1)=c, g(t2)=-c." is not necessarily true even for continuous functions.)
then this is an error in the Braun Springer text in the early chapters. I looked at this a LONG time but instantly thought how can he come up with g(t1)=c, g(t2)=-c from assuming g isn't constant. is this a really bad book? it covers a lot of nice applications and has some interesting problems
it bothered me so much I drank a beer and vomited. how can there be an error in this text????