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