when we say a function is continuous we mean that there are no breaks or holes.speaking mathematically consider a function f(x) say a point c is in its domain,then as the function approaches from both left and right of c it must be equal to the value at c  [f(c)].here since at all vaues of x the value is a constant therefore,it is continuous.

Function f(x) will be continuous at x=c if there is no break in the graph of function f(x) at the pt(x,f(x)).

In an interval it is said to be continuous if there is no break in the graph of function in the entire interval.

A function is continuous at x=c ifLim f(c) = f(c) = Lim f(c)

lim f(x) = f(c) = lim f(x)

x--->c-               x --->c+

So,for a constant function

for e.g. f(x)=k if we check continuity at any pt let it be x=c,

lim f(x) = k

x--->c-

f(c) = k

lim f(x) = k

x--->c+

here we see that left hand limit,right han limit an f(c) all are equal for all c belongs to R.

thereore, a constant function is continuous at all pts.