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A piecewise function is a function defined by two or more equations. A step functions is a piecewise function defined by a constant value over each part of its domain. You can write absolute value functions and step functions as piecewise functions so they're easier to graph.
A piecewise defined function is a function which is defined symbolically using two or more formulas
Assuming you mean "derivative", I believe it really depends on the function. In the general case, there is no guarantee that the first derivative is piecewise continuous, or that it is even defined.
The assertion is not true. Consider the function f(x) =|x - 3|, which is the distance of x from the point 3. The function is defined for all x but the absolute value is required.
Limits (or limiting values) are values that a function may approach (but not actually reach) as the argument of the function approaches some given value. The function is usually not defined for that particular value of the argument.