linear interpolation error estimate for non-smooth function

Question

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be bounded by the second derivative in $C$ norm of the function multiplied by $(x_2-x_1)^2$, i.e. $\sup_{x∈[x1,x2]}f^{′′}(x)(x_2-x_1)^2$ and some multiplying constant. If $f^{''}$ doesn't exist, can I replace the error estimate with $L^{\infty}$ norm for $f^{''}$, provided second weak derivative exist? How can I treat such an estimate? I sort of bounded the error and have it as a second order, but the constant is measured a.s. because of the weak derivative, so I should say that the error estimate holds almost everywhere?