Functional Equation Problem
Point of post: In this post I will answer the question
” What functions are continuous at and satisfy
Problem: Find all functions which are continuous at and satisfy
Proof: Note first that if
And so in particular, for any we have that
and thus by continuity at we see that
Otherwise, we note that for any and so we notice that
and thus by continuity
from where it follows that is a constant function.
1. Kaczor, Wiesĺawa J., and Maria T. Nowak. Problems in Mathematical Analysis. Providence (R.I.): American Mathematical Society, 2001. Print.
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