First of all, I would like to apologize if I make grammatical errors because English is not my mother tongue.
During a lecture at the university my teacher did an exercise that required to find the values of some parameters so that a function would be uniformly continuous.
The function has a bounded derivative in the whole domain regardless of the parameters except in a point of it.
To determine the value of the parameters, my professor has set two conditions:
1) The function must be continuous in the point.
2) The function must have a bounded derivative at that point.
After doing this he considered the exercise solved.
In fact, the bounded derivative implies that the function is Lipschitz continuous and this in turn implies that it is uniformly continuous.
My question is:
There are functions, such as $f(x) = \sqrt{x}$ defined in $[0,1]$ that are uniformly continuous but not Lipschitz continuous.
So how did my professor be sure he did not miss similar situations during the course of the exercise?
I would also like to ask you, in case my professor had forgotten about this, how could I find the values of the parameters omitted in his resolution of the exercise.
In case you need it, the function that my professor has examined is the following: $$\begin{cases}a\cdot\cosh x + b\cdot\cos x + \frac{b}{x-1}&x> 0\\ x + 3b(x-1) + \frac{2a}{x-1}& x \le 0\end{cases}$$
The parameters are $a$, $b$ and the solutions found by the professor are: $a = 0$ and $b = 1$.