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Inverse function theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the inverse function theorem is a <a href="/facts/Theorem/f2Pe1FMD">theorem</a> that asserts that, if a <a href="/facts/Real_function/bbNG3DnN">real function</a> <i>f</i> has a <a href="/facts/Continuously_differentiable_function/jQqTgLk1">continuous derivative</a> near a point where its derivative is nonzero, then, near this point, <i>f</i> has an <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a>. The inverse function is also <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a>, and the <i><a href="/facts/Inverse_function_rule/B2S9IIPQ">inverse function rule</a></i> expresses its derivative as the <a href="/facts/Multiplicative_inverse/yczdYyCw">multiplicative inverse</a> of the derivative of <i>f</i>. </p><p>The theorem applies verbatim to <a href="/facts/Complex-valued_function/hgCoQlH0">complex-valued functions</a> of a <a href="/facts/Complex_number/2w7mqI7e">complex variable</a>. It generalizes to functions from <i>n</i>-<a href="/facts/Tuples/BI1HIxL8">tuples</a> (of real or complex numbers) to <i>n</i>-tuples, and to functions between <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> of the same finite dimension, by replacing "derivative" with "<a href="/facts/Jacobian_matrix/0tNv6o8j">Jacobian matrix</a>" and "nonzero derivative" with "nonzero <a href="/facts/Jacobian_determinant/0tNv6o8j">Jacobian determinant</a>". </p><p>If the function of the theorem belongs to a higher <a href="/facts/Differentiability_class/mffL4ch2">differentiability class</a>, the same is true for the inverse function. There are also versions of the inverse function theorem for <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a>, for differentiable maps between <a href="/facts/Manifold/rXPERJtu">manifolds</a>, for differentiable functions between <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>, and so forth. </p><p>The theorem was first established by <a href="/facts/%25C3%2589mile_Picard/fq6m32JE">Picard</a> and <a href="/facts/%25C3%2589douard_Goursat/kMndmd26">Goursat</a> using an iterative scheme: the basic idea is to prove a <a href="/facts/Fixed_point_theorem/Jvkz4SUu">fixed point theorem</a> using the <a href="/facts/Contraction_mapping_theorem/TRygeoCP">contraction mapping theorem</a>. </p>