## 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

that

And so in particular, for any we have that

and thus by continuity at we see that

so 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.

**References:**

1. Kaczor, Wiesĺawa J., and Maria T. Nowak. *Problems in Mathematical Analysis.* Providence (R.I.): American Mathematical Society, 2001. Print.

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