只包含整数数字

用正则表达式即可实现。

var num = function(str) {
    var patrn = /^[0-9]{1,20}$/;
    var bool = true;
    if (!patrn.exec(str)) {
        bool = false;
    }
    console.log(bool);
    return bool;
}

num("3.4");
num("34");
num("");
num("  ");
num("0-34");
num(">");

得到的结果：

false
true
false
false
false
false

包含带小数点的数字

若使用上面的正则匹配，会将小数点识别为非数字字符。可利用 Number() 将字符串强制转换成数字，若结果不为 NaN，则说明该字符串只包含数字。
但是 Number() 会将空字符串与空格都转换为 0，所以要先将空字符串与空格排除。

var strToNum = function(str) {
    var convertNum = Number(str); // 将字符串强制转换为数字
    if (str === "") { // 排除空字符串
        console.log("\"" + str + "\" is an Empty String.");
    } else {
        if (str.includes(" ")) { // 排除空格
            console.log("\"" + str + "\" contains spaces.");
        } else {
            if (isNaN(convertNum)) {
                console.log("\"" + str + "\" can't be converted into a number.");
            } else {
                console.log("\"" + str + "\" can be converted into " + convertNum + ".");
            }
        }
    }
}

// 测试函数
strToNum("34");
strToNum("0.34");
strToNum(",");
strToNum(" ");
strToNum("");
strToNum("  89 ");
strToNum("2.34");
strToNum("2-34");

得到的结果：

"34" can be converted into 34.
"0.34" can be converted into 0.34.
"," can't be converted into a number.
" " contains spaces.
"" is an Empty String.
"  89 " contains spaces.
"2.34" can be converted into 2.34.
"2-34" can't be converted into a number.