Easy counterexamples have been mentioned. What is true, however, is the fact that sum of a differentiable function and a nondifferentiable function is nondifferentiable. This is true because differentiable functions are closed under subtraction, i.e. they comprise a subgroup of all functions. Hence the claimed property is simply a special case of the following complementary form of the subgroup property from my prior post.
THEOREM $\ $ A nonempty subset $\rm\:S\:$ of abelian group $\rm\:G\:$
comprises a subgroup $\rm\iff\ S\ + \ \bar S\ =\ \bar S\ $ where $\rm\: \bar S\:$ is the complement of $\rm\:S\:$ in $\rm\:G$
Instances of this are ubiquitous in concrete number systems, e.g.
algebraic * nonalgebraic = nonalgebraic if nonzero
rational * irrrational = irrational if nonzero
real * nonreal = nonreal if nonzero
even + odd = odd additive example
integer + noninteger = noninteger