Let (X_n,Y_n)_n≤1 be a R^dR valued stationary process. Define the estimator of the conditional mode of Y₁ given X₁=x as the random variable θ_n(x) that maximizes a kernel estimator of the conditional density of Y₁ given X₁ = x. We establish asymptotic normality of θ_n(x) when the process (X_n,Y_n)_n≤1 is assumed to be strongly mixing. We derive from our results asymptotic normality of a predictor and propose a confidence bands for the conditional mode function. A simulation study shows how good the normality of the conditional mode function estimator is when dealing with samples of finite sizes.