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Sigma-martingales are covered in Philip Protters' book: Stochastic Integral and Differential Equations. In the second edition (Vers 2.1) Chapter IV.9 treats this topic.
Corollary 2 of this chapter establishes that a local martingale is a sigma martingale.
Theorem 91 establishes that a sigma martingale which is a special semimartingale is a local martingale. Since continuous semimartingales are special, it follows that any continuous sigma-martingale is a local martingale.
The famous Emery example is given directly before Theorem 34 - it is a badly behaving stochastic integral and the standard example of a sigma martingale which is not a local martingale (but with jumps): consider $X=U \, \mathbf{1}_{t \ge T}$ where $P(U=1)=P(U=-1)=1/2$, and $T$ is exponential with intensity $1$. Then $X$ is a martingale (in its own filtration). Now choose $H_t=\frac 1 t \, \mathbf{1}_{t >0}$. Then the stochastic integral $Z=H \cdot X$ is not a local martingale because $E[|Z_\tau|]= \infty$ for any stopping time $\tau$ with $P(\tau >0)>0$.