So, I have the following, for a real valued random variable $K$ with a pdf $f$ there is a real valued mean $\mu$ s.t.

$f(\mu +k) = f(\mu - k)$, and I need to show that for some random variables $(\mu - K)$ and $(K - \mu)$, their c.d.f. are the same.

From what I understand, I have a pdf for a symmetric distribution around the mean.

Now, let $Y = (\mu - K)$, then $(K - \mu) = -(\mu - K) = -Y$, and Basically, I need to show that $F(Y) = F(-Y)$.

But, by definition of cumulative distribution function, $F(Y) = P(Y \leq y) = \int_{-\infty}^{y}f(y)dy$, while $F(-Y) = P(Y \leq -y) = \int_{-\infty}^{-y}f(-y)dy$ and from how I see it $F(-Y) < F(Y)$.

In fact I can define $F(Y)$ in this way, $F(Y) = P(Y \leq y) = P(-\infty \leq Y \leq -y) + P(-y < Y \leq y)$

Could you point out what I get wrong?