Wave function is continuous???

Let me be blunt here guys, my question is Why should the wave function have to be continuous??

If the potential is continuous in a region then the wavefunction has to be continous in that region otherwise second derivative of wavefunction is not defined hence schrodinger is not solvable.

can u please tell me why the 2nd derivative of the wave function is not defined

Since if a function is not continuous then its first derivative does not exist at pt. of discontinuity.
Hence second derivative can also not exist.
Refer to the image in the link.

https://www.google.co.in/imgres?imgurl=http%3A%2F%2Fcdn-0.analyzemath.com%2Fcalculus%2Fcontinuity%2Fcontinuous_3.gif&imgrefurl=http%3A%2F%2Fwww.analyzemath.com%2Fcalculus%2Fcontinuity%2Fcontinuous_functions.html&docid=FbUbjrRCSjnxQM&tbnid=8BhjJKVwbN5bkM%3A&w=293&h=224&bih=662&biw=1366&ved=0ahUKEwjPy_q-kYDPAhXGro8KHdEABc8QMwgkKAcwBw&iact=mrc&uact=8

i know i'm moving away from the discussion but a quick question if the limit of a function exists at a point then is the fuction differentiable at that point

No , continuity is a necessary condition for differentiability.Check this https://en.wikipedia.org/wiki/Differentiable_function

Differentiability and continuity

okay then lets think in a non mathematical manner. Say the wave function is not continuous at a point, then how do you think you can explain the non continuous distribution of the probability. And then anyways the potential energy in the box is uniform.