👉 In mathematics, the corresponding theorem states that if a function \( f \) is defined on an interval \( I \) and maps \( I \) to a subset \( f(I) \), then for every \( y \) in \( f(I) \), there exists a unique \( x \) in \( I \) such that \( f(x) = y \). This is known as the Inverse Function Theorem. It essentially guarantees that if a function is continuous and differentiable on an interval, it has a well-defined inverse function on the corresponding image set, and this inverse is unique. This theorem is crucial in understanding the behavior of functions, particularly in areas like calculus and analysis, where it helps establish the existence and uniqueness of solutions to equations involving functions.
