Because I am not majoring in math, I wonder if there is a standard approach to prove or disprove Lipschitz continuity.
In my case, I want to prove that the Mean Squared Error (MSE) loss function for the weight ( w ) is Lipschitz continuous.
MSE is defined as: $$ E_{in}(w) = \frac{1}{n}\sum_{i=1}^{n}(y_i - w^Tx_i)^2 $$
As we know, in particular, a real-valued function $f: \mathbb{R}^d → \mathbb{R}$ is called Lipschitz continuous if there exists a positive real constant K such that, for all real $x_1$ and $x_2$, $$ |f(x_1) - f(x_2)| \leq K \|x_1 - x_2\|_2 $$
I tried to analyze $|E_{in}(w_1) - E_{in}(w_2)|$, but the result involves $ w_1$ and $w_2$, so I couldn't find a constant $K$ to satisfy the condition.
However, this observation alone is not sufficient to disprove that MSE is Lipschitz continuous. Do I need to provide a counterexample to disprove it?
If a counterexample is necessary, does it mean that I should check some simpler conditions to test satisfaction before attempting a proof?
As commentators noted and shown below, a function of the form $x^2$ or equivalent such as the one in the MSE above is not Lipschitz.
For your counter-example, consider $d=1$ i.e. $\omega$ is one dimensional
$|\frac{1}{n}\sum_{i=1}^{n}[(y_i-\omega_1x_i)^2-(y_i-\omega_2x_i)^2]|=|\frac{1}{n}\sum_{i=1}^{n}[y_i^2-2y_i\omega_1x_i+\omega_1^2x_i^2-y_i^2+2y_i\omega_2x_i-\omega_2^2x_i^2]|$
Now suppose, $\omega_1=M,\omega_2=-M$
$|\frac{1}{n}\sum_{i=1}^{n}[(y_i-\omega_1x_i)^2-(y_i-\omega_2x_i)^2]|=|\frac{1}{n}\sum_{i=1}^{n}[-2y_iMx_i+M^2x_i^2-y_i^2-2y_iMx_i-Mx_i^2]|=\\|\frac{1}{n}\sum_{i=1}^{n}-4y_iMx_i|$
Which so long as not all the data is exactly 0 shows that this difference will grow to infinity for $M$ large.