I want to know if there is a name for this property: for given positive numbers $a,b$, we have an inequality $|f(x)-f(y)|<b$ whenever $|x-y|<a$.
Warning:
It looks like Lipshitz continuity, but this property does not imply the continuity; for example, EVERY bounded function with bound $M$ satisfies it when we set $a$ and $b$ such that $a>0$ and $b>2M$. Note that this property depends on the choice of $a$ and $b$. Maybe the existence of these constants is important.
Where did I see this?:
I just saw this in several exam problems. I thought it might be useful in numerical analysis, because making $b$ small means that this function can be considered to be a Lipschitz continuous function with small errors. So I guessed it should have a name.
If anyone knows any concept similar to this, please tell me. Thanks.
Not all properties are required to have names, we can just make up a name like $P(a,b)$. However, your property is similar to the concept of a "leaky bucket constrained" arrival process used in queueing theory.
Definition 1:
Given positive real numbers $a,b$, a function $f:\mathbb{R}\rightarrow\mathbb{R}$ has property P(a,b) if $|f(x)-f(y)|\leq b$ whenever $|x-y|\leq a$.
Definition 2:
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is leaky bucket constrained with rate parameter $\lambda \geq 0$ and burst parameter $\sigma \geq 0$ (written $f(t) \sim leaky(\lambda, \sigma)$) if $$ |f(x)-f(y)|\leq \lambda |x-y| + \sigma \quad \forall x, y \in \mathbb{R}$$
From these two definitions we can prove the following:
$f(t) \sim leaky(\lambda, \sigma) \implies$ $f(t)$ has property $P(a,\lambda a+\sigma)$ for every $a\geq 0$.
If $f(t)$ has property $P(a,b)$ for some positive numbers $a,b$ then $f(t) \sim leaky(b/a, b)$.
Definition 2 is usually applied to nondecreasing functions that represent arrival functions, such as functions $N(t)$ that represent the number of jobs that arrive to a queue during a time interval $[0,t]$. Such $N(t)$ functions jump discontinuously when a new arrival occurs. If $N(t) \sim leaky(\lambda, \sigma)$ then any interval of size $T$ has at most $\lambda T + \sigma$ arrivals, and the long term arrival rate satisfies $$\limsup_{t\rightarrow\infty} \frac{N(t)}{t} \leq \lambda$$ Such processes are useful because, assuming each job has a fixed size, the worst-case size of the queue can be deterministically bounded (in terms of $\sigma$) whenever the service time of each job is a constant $1/\mu$ that satisfies $\mu\geq \lambda$.