In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. An example is given by the Heaviside step function. The term piecewise is also used as an adverb to describe a property of a piecewise function that holds on each interval but may not hold on the whole function. Euclid, detail from The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... An expression in the very basic sense is the noun form of the verb express. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

A piecewise-continuous function given by two different quadratic functions to the left and to the right of x₀.

According to the standard definitions, this is a single function, that happens to have its value computed by different methods in different cases. It is useful to do this, for example to make a sawtooth function. That is an example of a piecewise linear function: its graph is made up of a number of parts of the graphs of linear functions. Problems can arise at the ends of the intervals used for separate definitions. We must give a definite value for f(x) there, as everywhere else. It may be a point where continuity fails (as for the Heaviside function at 0), or where the function isn't smooth (the absolute value function at 0). Image File history File links Upper_semi. ... Image File history File links Upper_semi. ... f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where is nonzero. ... The sawtooth wave is a kind of basic waveform. ... In mathematics, a piecewise linear function , where V is a vector space and is a subset of a vector space, is any function with the property that can be decomposed into finitely many convex polytopes, such that f is equal to a linear function on each of these polytopes. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

The definitions of piecewise continuous, piecewise differentiable and so on are therefore made, to require that the 'pieces' of the function are continuous (resp. differentiable), but that at the end points failure of those conditions is allowed. A path said to be piecewise continuously differentiable is a continuous path (in the plane, say) but which can at some points turn direction sharply, so the continuity of the derivative vector at those points doesn't hold. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Look up path in Wiktionary, the free dictionary. ...

See also

• Spline
• B-spline