Question

For \[x\ne c\], which identity is used to prove that differentiability implies continuity?

Options

• \[f(x)-f(c)=f'(c)(x-c)^2\]

• \[f(x)-f(c)=\frac{f(x)-f(c)}{x-c}+(x-c)\]

• \[f(x)-f(c)=\frac{x-c}{f(x)-f(c)}\]

• \[f(x)-f(c)=\frac{f(x)-f(c)}{x-c}\cdot(x-c)\]

Answer 1

For \[x\ne c\], multiplying the difference quotient by \[x-c\] recovers \[f(x)-f(c)\]. This factorization allows the limit to be evaluated as a product.