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Multi-beam LiDAR sensors are increasingly used in robotics, particularly for autonomous cars for localization and perception tasks. However, perception is closely linked to the localization task and the robot's ability to build a fine map of its environment. For this, we propose a new real-time LiDAR odometry method called CT-ICP, as well as a complete SLAM with loop closure. The principle of CT-ICP is to use an elastic formulation of the trajectory, with a continuity of poses intrascan and discontinuity between scans, to be more robust to high frequencies in the movements of the sensor. The registration is based on scan-to-map with a dense point cloud as map structured in sparse voxels to operate in real time. At the same time, a fast method of loop closure detection using elevation images and an optimization of poses by graph allows to obtain a complete SLAM purely on LiDAR. To show the robustness of the method, we tested it on seven datasets: KITTI, KITTIraw, KITTI-360, KITTI-CARLA, ParisLuco, Newer College, and NCLT in driving and high-frequency motion scenarios. The CT-ICP odometry is implemented in C++ and available online. The loop detection and pose graph optimization is in the framework pyLiDAR-SLAM in Python and also available online. CT-ICP is currently first, among those giving access to a public code, on the KITTI odometry leaderboard, with an average Relative Translation Error (RTE) of 0.59% and an average time per scan of 60ms on a CPU with a single thread.