Let $A$ be a bounded self-adjoint linear operator on a Hilbert space $H$. For any analytic function $f$ whose radius of convergence contains the spectral radius of $A$ we may compute the functional calculus as follows $$f(x) = \sum a_n x^n \implies f(A) = \sum a_n A^n.$$ Using the above, the fact that $A$ is self-adjoint, and the Weisrestress approximation theorem, we may extend the above to the continuous functions, and then further to the set of all Borel measurable functions.
In the analytic case it is easy to compute the functional calculus as above, but how is this done explicitly for continuous or Borel functions that are not analytic? That is, if $f$ is a Borel function and we have an explicit form for $A$, how do we compute $f(A)$? Is this always possible?