A greeting bot for a colleague from work
JavaScript (ES6), 43 bytes

f=(D,t,d)=&gt;5&gt;&gt;d&amp;t&gt;749&amp;t&lt;811&amp;&amp;!f[D]*(f[D]=1)

Try it online!
Input

the date as a string in yyyymmdd format
the time as a string in hhmm format
the day of week as a 0-indexed integer, with 0 = Tuesday, 1 = Wednesday, ..., 6 = Monday

Output
Returns 0 or 1.
Commented
f = (            // named function, as the underlying object will be used as storage
  D,             // D = date (string)
  t,             // t = time (string)
  d              // d = day of week (integer)
) =&gt;             //
  5              // 5 is 0000101 in binary, where 1's are set for Tuesday and Thursday
  &gt;&gt; d &amp;         // test the relevant bit for the requested day of week
  t &gt; 749 &amp;      // test whether we are in the correct time slot
  t &lt; 811        //
  &amp;&amp; !f[D] *     // make sure that this date was not already invoked at a correct time
  (f[D] = 1)     // and store it in the underlying object of f()