Some important properties of continuous functions:
If the functions f(x) and g(x) are both continuous at x =a then the following results hold true:
1. cf (x) is continuous at x =a where c is any constant.
2. f(x) + g(x) is continuous at x = a.
3. f(x).g(x) is continuous at x= a.
4. f(x)/g(x) is continuous at x= a, provided g(a) ≠ 0.
Functions f(x)  Interval in which f(x) is continuous 
Constant C  (∞,∞) 
bn, n is an integer > 0  (∞,∞) 
xa  (∞,∞) 
x^{n}, n is a positive integer.  (∞,∞){0} 
a_{0}x^{n} + a_{1}x^{n1 }+........ + a_{n1}x + a_{n}  (∞,∞) 
p(x)/q(x), p(x) and q(x) are polynomials in x  R  {x:q(x)=0} 
sin x  R 
cos x  R 
tan x  R{nπ:n=0,±1,........} 
cot x  R{(2n1)π/2:n=0,±1,± 2,........} 
sec x  R{(2n1)π/2:n=0,±1,± 2,........} 
e^{x}  R 
ln x  (0, ∞) 
If you know the graph of a function, it can be easily judged without even solving whether a function is continuous or not. The graph below clearly shows that the function is discontinuous.
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