In mathematics, a real analytic function is a real function of a real variable, which is defined and continuous for every real value, which possesses derivatives (also real analytic) of all orders, and such that the Taylor series expansion around every point is valid in some neighborhood of that point. An example is the exponential function exp(x). However, the function defined as for nonzero values and zero when x is zero is not real analytic, despite having derivatives of all orders, since the Taylor series around zero does not give the correct value in any neighborhood of zero, being identically zero. A real analytic function may not be a complex analytic function; for example has no real singularities, but is singular at i and -i.