Prove that the European call payoff function is Lipschitz continuous, however, the payoff of the digital option $\mathbb{1}_{\{S_T > K\}}$ is not Lipschitz continuous.
The European call payoff function is defined as $$f(S_T) = \max\{S_T - K, 0\}$$ where $K$ is the strike price and $T$ is the time to maturity. I am not sure how to prove this result, any suggestions would be greatly appreciated.
Let $f(x)=(x-K)^+$, and consider the following three cases:
If $x,y<K$ then $f(x)=f(y)=0$, so $|f(x)-f(y)|=0$. If $x,y\geq K$ then $$ |f(x)-f(y)|=|x-K-(y-K)|=|x-y|$$ Finally, if say $x<K$ but $y\geq K$ then $$ |f(x)-f(y)|=y-K\leq y-x=|x-y| $$
So in all three cases $|f(x)-f(y)|\leq |x-y|$, so $f$ is Lipschitz.
For the second question, note that $g(x)=1_{x>K}$ is discontinuous at $x=K$.