2D jumping not working

Unity Engine
1

Hello im still new to unity development and im trying to get the main character to jump and the best i can get is it will jump if and only if i hold down the jump button, and if i don’t instead my y axis changes to wherever i am at when i lift my finger off the jump button, so if i spam the jump button i just climb and leave the level. I have my jump animation through the actual animation clip. Here is the code.

using UnityEngine;

public class CharacterMovement : MonoBehaviour {
    // RidgidBody component instance for the player
    private Rigidbody2D playerRigidBody2D;

    // Variable to track how much movement is needed from input
    private float movePlayerVector;

    // Referencing to the player's animator component.
    private Animator anim;

    // For determining which way the player is currently facing.
    private bool facingRight;

    // For determining if the jump is activated
    private bool characterJump;

    // Speed modifier for player movement
    public float speed = 4.0f;

    // Reference to the player's sprite GameObject
    private GameObject playerSprite;

    // Initialize any component references
    void Awake() {
        playerRigidBody2D = (Rigidbody2D)GetComponent(typeof(Rigidbody2D));
        playerSprite = transform.Find("PlayerSprite").gameObject;
        anim = (Animator)playerSprite.GetComponent(typeof(Animator));

    }

    // Update is called once per frame
    void Update() {
        // Cache the horizontal input.
        movePlayerVector = Input.GetAxis("Horizontal");

        characterJump = Input.GetButton("Jump");

        anim.SetBool("jump", characterJump);

        anim.SetFloat("speed", Mathf.Abs(movePlayerVector));

        playerRigidBody2D.velocity = new Vector2(movePlayerVector * speed, playerRigidBody2D.velocity.y);

        if (movePlayerVector > 0 && !facingRight) {
            Flip();
        }
        else if (movePlayerVector < 0 && facingRight) {
            Flip();
        }

    }

    void Flip() {
        // Switch the way the player is labeled as facing.
        facingRight = !facingRight;

        // Multiply the player's x local scale by -1.
        Vector3 theScale = playerSprite.transform.localScale;
        theScale.x *= -1;
        playerSprite.transform.localScale = theScale;

    }

}
2

You should check if your player is on the ground only then can the player jump.
A Good tutorial is gamesplusjames on YouTube