@dm63 has demonstrated in his answer that without additional constraint, the delta can exceed $1$. Analogously, in the continuous time and stock price setting, assume the volatility of stock price below a positive threshold vanishes and is a positive constant above that threshold. Set the European call strike at the threshold. We see that the current European call price is zero when the current price is right below the threshold and positive right above the threshold. So the delta at the threshold is infinity.

With additional smoothness and dependency constraints on the volatility, however, the European call delta will be no greater than $1$. I shall supply such a condition and the associated proof.

@dm63 has demonstrated in his answer that without additional constraint, the delta in discrete setting can exceed $1$. Analogously, in the continuous time and stock price setting, assume the volatility of stock price below a positive threshold vanishes and is a positive constant above that threshold. Set the European call strike at the threshold. We see that the current European call price is zero when the current price is right below the threshold and positive right above the threshold. So the delta at the threshold is infinity.

With additional smoothness and dependency constraints on the volatility, however, the European call delta will be no greater than $1$. I shall supply such a condition and the associated proof.