Defining one variable function on specific interval and taking supremum of it

I got a course in my undergraduate education 4 years ago. So actually I can’t remember the details about mathematica. I’m trying to solve one thing about 1 week. Here is my question:

f is a continuos and bounded function on [a,b]. Define

blank”>{x,t\in[a,b]&space;and&space;|t-x|<\delta}|f(x)-f(t)|” title=”w(f,\delta)=sup_{x,t\in[a,b] and |t-x|<\delta}|f(x)-f(t)|” />

I want to write this function on mathematica and checking while delta going zero, is w will be going with it. I replaced supremum with MaxValue function. But I didn’t write when x,t in [a,b] take f(x)-f(t). Because f is a single variable and I dont know how can I put condition when t-x < delta.

These are my problems. After defining this function I want to see on graph how it changes with different delta’s