Linear interpolation ($(1-t)f(a)+tf(b)$)

Between two prescribed values, linear interpolation assigns $(1-t)f(a)+tf(b)$ at $(1-t)a+tb$, for $0\le t\le 1$. Joining successive samples produces a piecewise linear function. For a continuous function on a compact interval, interpolants along grids with mesh tending to zero converge in the uniform norm; their error is bounded by the modulus of continuity at the mesh size.

A linear interpolant is the unique polynomial of polynomial degree at most one taking prescribed values $y_a,y_b$ at distinct interpolation nodes $a,b$. It is an affine function, including the constant function case. For $a\le x\le b$, its two displayed coefficients are nonnegative and sum to one, so changing each sampled value by at most $\delta$ changes every interpolated value by at most $\delta$. Differentiation instead uses the difference of the two samples divided by $b-a$, causing noise amplification by differentiation of a linear interpolant.