This concept applies the Mean Value Theorem (MVT) to prove that a continuous, differentiable function with a nowhere-zero derivative is injective (one-to-one): for any two distinct inputs, MVT guarantees a point c between them where the secant slope equals f′(c), and since f′(c) is nonzero and the input difference is nonzero, their product — the output difference — must also be nonzero. This belongs to single-variable calculus, specifically the theory of using derivative sign/nonvanishing conditions to deduce global monotonicity or injectivity properties of a function, paralleling the related MVT-based proof that a positive (or negative) derivative implies strict monotonicity.
This concept applies the Mean Value Theorem (MVT) to prove that a continuous, differentiable function with a nowhere-zero derivative is injective (one-to-one): for any two distinct inputs, MVT guaran…