# Find all continuous functions with the following properties.

How do I find all continuous functions $f:[a,b]\to \mathbb{R}$ such that $$\int_{a}^x f(t) dt = \int_{x}^b f(t) dt$$ , for all $x \in (a,b)$

This problem is from old qualifyings of real analysis , it does not look too bad but still I struggled quit a good time. I tried to define some functions to work out but non of them did the job.

Any little hint would be appreciated.

• Taking a derivative might help. – muaddib Jun 1 '15 at 23:41
• do not we need $f$ to be differentiable? – mp100 Jun 1 '15 at 23:42
• Derivative of the integral <--> Fundamental Theorem of Calculus – muaddib Jun 1 '15 at 23:43
• @mp100 Not to differentiate $F\colon x\mapsto =\int_a^x f$, which is differentiable (even $C^1$). – Clement C. Jun 1 '15 at 23:43
• The only function satisfying that requirement I got is a zero function. I do not think that we can have such non zero function for arbitrary $x$. – mp100 Jun 2 '15 at 0:04

## 2 Answers

Hint: $\int_a^x = \int_a^b - \int_x^b = \ldots$.

If a = b, then any function works. If a < b, then f is a constant function equal to 0. Really, for any c < d from [a, b] consider three integrals:

I1 from a to c, I2 from c to d, I3 from d to b.

Obviously I1 + I2 = I3 and I1 = I2 + I3 by the property of f. This means I2 is 0. Because it is true for any c, d, integral from a to x is always 0 for any x from [a, b]. Taking its derivative by x becomes 0, and it is also f.