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We present a convolutional neural network that is capable of searching for continuous gravitational waves, quasi-monochromatic, persistent signals arising from asymmetrically rotating neutron stars, in $\sim 1$ year of simulated data that is plagued by non-stationary, narrow-band disturbances, i.e., lines. Our network has learned to classify the input strain data into four categories: (1) only Gaussian noise, (2) an astrophysical signal injected into Gaussian noise, (3) a line embedded in Gaussian noise, and (4) an astrophysical signal contaminated by both Gaussian noise and line noise. In our algorithm, different frequencies are treated independently; therefore, our network is robust against sets of evenly-spaced lines, i.e., combs, and we only need to consider perfectly sinusoidal line in this work. We find that our neural network can distinguish between astrophysical signals and lines with high accuracy. In a frequency band without line noise, the sensitivity depth of our network is about $\mathcal{D}^{95\%} \simeq 43.9$ with a false alarm probability of $\sim 0.5\%$, while in the presence of line noise, we can maintain a false alarm probability of $\sim 10\%$ and achieve $\mathcal{D}^\mathrm{95\%} \simeq 3.62$ when the line noise amplitude is $h_0^\mathrm{line}/\sqrt{S_\mathrm{n}(f_k)} = 1.0$. We evaluate the computational cost of our method to be $O(10^{19})$ floating point operations, and compare it to those from standard all-sky searches, putting aside differences between covered parameter spaces. Our results show that our method is more efficient by one or two orders of magnitude than standard searches. Although our neural network takes about $O(10^8)$ sec to employ using our current facilities (a single GPU of GTX1080Ti), we expect that it can be reduced to an acceptable level by utilizing a larger number of improved GPUs.