How big is the discontinuity? I'm assuming the two waveforms are both essentially zero-mean waveforms. If not, you'll want to shift them to make them zero mean.
Then, one thing you can do is to apply a moving average filter to smooth out the discontinuity. I'll use a 5-point moving average filter here. I would try a low order filter first, because you probably don't want to filter your signal too much (it is a lowpass filter). The larger the magnitude of the discontinuity, the higher order you would need.
t = 0:0.01:1;
x = cos(2*pi*t).*(t<=0.5)+cos(2*pi*t-pi/8).*(t>0.5);
plot(t,x,'r'); hold on;
h = 1/5*ones(5,1);
y = filtfilt(h,1,x);
plot(t,y,'b')
