the simplest examples of uniformly continuous functions. Any continuous function on the interval is also uniformly continuous, since is a compact set. Every function which is differentiable and has bounded derivative is uniformly continuous. Every Lipschitz continuous map between two metric spaces is uniformly continuous. More generally, every Hölder continuous function is uniformly continuous. The absolute value function is uniformly continuous, despite not being differentiable at . This shows uniformly continuous functions are not always differentiable. Despite being nowhere differentiable, the Weierstrass function is uniformly continuous. Every member of a uniformly equicontinuous set of functions is uniformly continuous. Nonexamples Functions that are unbounded on a bounded domain are not uniformly continuous. The tangent function is continuous on the interval but is not uniformly continuous on that interval, as it goes to infinity as . Functions that have slopes that become unbounded on an infinite domain cannot be uniformly continuous. The exponential function is continuous everywhere on the real line but is not uniformly continuous on the line, since its derivative tends to infinity as . Visualization For a uniformly continuous function, for every positive real number there is a positive real number such that two function values and have the maximum distance whenever and are within the maximum distance . Thus at each point of the graph, if we draw a rectangle with a height slightly less than and width a slightly less than around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle. For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.) History The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. The proofs are almost verbatim given by Dirichlet in his lectures on definite integrals in 1854. The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous.