Does $f(t,y)=4t\sqrt{y}$ satisfy the Lipschitz Condition?

Question

$f(t,y)=4t\sqrt{y}$.

Does $f$ satisfy the Lipschitz Condition?

In other words how do I check if $$|f(t, y_1) - f(t, y_2)| = |4t\sqrt{y_1} - 4t\sqrt{y_2}| \le C |y_1- y_2|$$ holds.

EDIT: My Domain is $0<t<b$ for some real positive constant $b$ and $0<y<\infty$

Answer 1

You have not specified the domain. $|\sqrt {y_1} -\sqrt {y_2}| \leq C|y_1-y_2|$ does not hold for $y_1$ and $y_2$ near $0$. Reason: for $y_1,y_2 >0$ we have $|y_1-y_2|=|\sqrt {y_1} -\sqrt {y_2}| |\sqrt {y_1} +\sqrt {y_2}|$. The inequality becomes $1 \leq C|\sqrt {y_1} +\sqrt {y_2}|$ which leads to a contradiction if you let $y_1,y_2 \to 0$.

Answer 2

By use of the binomial theorem you get $$ |f(t,y_1)-f(t,y_2)|= 4|t|\frac{|y_1-y_2|}{\sqrt{y_1}+\sqrt{y_2}}. $$ This, as has been observed multiple times, is unbounded if $y_1$ and $y_2$ are close to zero.