Given function (x) = x³ + x ² − 1 is polynomial of degree 3. 

Therefore, given function (x) = x ³ + x ² − 1 is defined for every real value of . 

Let us the continuity of function (x) = x ³ + x ² − 1 at arbitrary point x = c. 

The left hand limit of function at x = c is (c⁻) = \(\lim \limits _{x \to c^+}\) f(x) = \(\lim \limits _{h \to 0}\) f(c − ℎ) 

= \(\lim \limits _{h \to 0}\) (c − ℎ)³ + (c − h)² − 1 

= (c − 0)³ + (c − 0)² − 1 = c³ + c² − 1. (By putting the limit ) 

The right hand limit of function at x = c is f(c⁺) 

= \(\lim \limits _{x \to c^+}\) f(x) =\(\lim \limits _{h \to 0}\) f(c + h) 

= \(\lim \limits _{h \to 0}\) (c + h)³ + (c + h)² − 1 = (c + 0)³ + (c + 0)² − 1 = c³ + c² − 1. (By putting the limit ) 

And the functional value of function f(x) = x³ + x² − 1at x = c is f(c) = c³ + c² − 1. 

Hence, the left hand limit of function at x = c = the right hand limit of function at x = c = f(c). 

i.e., f(c⁻) = f(c ⁺) = f(c) = c³ + c² − 1. 

Therefore, the function f(x) =x³+ x² − 1is continuous at x = c, 

which is arbitrary point. 

Hence, f is continuous for each x ∈ R.