xcosx is non periodic why?

For a function to be periodic ,

f(x)=f(x+T)      Where T is the period of the function

for ex:    sinx=sin(x+2∏)      where 2∏ is the period

for any function to be periodic f(x)=f(x+t)

where t is period 

coming to question xcosx 

if u replce x by x+t u get (x+t)cos(x+t)

xcos(x+t)+tcos(x+t)=xcosx

this is possible only if t=0 therefore no period 

therefore given function is non periodic