I am reading diffusion process from a textbook and noticed that the author claims that condition (1) implies the stochastic process $X_{t}$ cannot have instantaneous jumps. So I wonder does "cannot have instantaneous jumps" implies $X_t$ is continuous in $t$? And why is it true? Moreover, can I show it using $\epsilon-\delta$ language?
Definition 10.8.3. An $\mathbb{R}^{n}$-valued Markov process $X_{t}, a \leq t \leq b$, is called a diffusion process if its transition probabilities $\left\{P_{s, x}(t, \cdot)\right\}$ satisfy the following three conditions for any $t \in[a, b], x \in \mathbb{R}^{n}$, and $c>0$ :
(1) $\lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon} \int_{|y-x| \geq c} P_{t, x}(t+\varepsilon, d y)=0$.
(2) $\lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon} \int_{|y-x|<c}\left(y_{i}-x_{i}\right) P_{t, x}(t+\varepsilon, d y)=\rho_{i}(t, x)$ exists.
(3) $\lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon} \int_{|y-x|<c}\left(y_{i}-x_{i}\right)\left(y_{j}-x_{j}\right) P_{t, x}(t+\varepsilon, d y)=Q_{i j}(t, x)$ exists.
Remark 10.8.4. Condition (1) implies that the stochastic process $X_{t}$ cannot have instantaneous jumps.