Smooth is not the same as analytic: the ladder of differentiability
In analysis, smoothness counts how many times a function can be differentiated before something breaks. The absolute value function is continuous but has a sharp corner, so it fails at the first step, while some functions can be differentiated forever yet still refuse to match their own Taylor series.
The measure is the differentiability class, a non-negative integer k. A function belongs to class Ck when its derivatives up to order k all exist and are continuous. Class C0 holds every continuous function; C1 holds the continuously differentiable ones, whose derivative exists and is itself continuous. A function with derivatives of every order is called infinitely differentiable, or smooth, though in practice smooth can also just mean differentiable enough for the problem at hand.
Each class sits strictly inside the one below. A function equal to x for non-negative inputs and zero otherwise is continuous but has no derivative at zero, so it is C0 but not C1. Stranger cases exist: x squared times the sine of one over x is differentiable everywhere, yet its derivative oscillates near zero, and swapping the square for the power four-thirds gives a derivative that is unbounded on any closed interval containing zero.
Above smoothness sits analyticity, class C omega, where a function's Taylor series converges to the function near every point. Trigonometric functions qualify wherever they are defined. Not every smooth function makes the cut, though. Bump functions, such as e raised to minus one over one minus x squared inside the interval from minus 1 to 1 and zero outside it, are infinitely differentiable yet fail to be analytic at the endpoints, and they vanish outside a bounded region.
These classes matter well beyond definitions. They describe degrees of regularity for partial differential equations and define kinds of differentiable manifolds in differential topology. Complex functions add a twist: treated as maps between real vector spaces they can have any class, but a function that is complex differentiable on an open set is automatically holomorphic, and therefore analytic, there.
Source: Smoothness
