I think this question is easy. However, I have not been able to solve it.
Let $a,\sigma:\mathbb{R}\times \mathbb R\to\mathbb{R}$, smooth functions such that $\sigma>0$. Consider the 1-dimensional SDE, $$dX_t = a(X_t,t) dt + \sigma(X_t,t) dW_t$$ $$X_0 = x_0\in\mathbb{R}. $$ where $W_t$ is a Brownian motion.
Fixing $y\in\mathbb R$ and $t>0$, I was interested in showing that$$\mathbb{P}\left(\{\omega \in \Omega;\ X_t = y\}\right)=0.$$ Where $(\Omega,\mathcal F, \mathbb P)$, is the probability space being considered.
Does anyone know if the above equation is true? A reference would be enough for me.