Show that the function f defined by
`f(x)={(x if "x is rational"),(0 if "x is irrational"):}`
is discontinuous everywhere except at x=0.

Show that the function f defined by f(x)={(1 if "x is rational"),(0if "x is irrational"):} is discontinuous AAne0inalpha

Show that the function f defined by f(x)={(x if "x is rational"),(-x if "x is irrational"):} is continuous at x=0 AAxne0inR

Show that the function f(x) given by f(x)={:{(x" sin "(1)/(x)", if "x!=0),(0" , if "x=0):} is continuous at x =0

Find the value of a such that the function f defined by f(x)={{:((sinax)/(sinx), "if", x ne 0),((1)/(a), "if", x=0):} is continuous at x=0

Show that the function f defined by f(x)=|1-x+|x|| , where x is any real number is a continuous function.

Is the function f defined by f(x)={(x", if "xle1),(5", if "xgt1):} continuous at x=0, at x=1 and at x=2 ?

Show that the function f(x)=2x-|x| is continuous at x=0 .

Show that the function f : R^+toR^+ defined by f (x) =x+1/x is injective, but not surjective .

Find the value of a such that the function f defined by f(x)={((sinax)/(sinx) if xne0),(1/a if x=0):} is continuous at x=0.

Show that the function f(x)=1/x is decreasing in (0,oo) .