section_path: Uses/Continuity of a function at a point
section_name: Continuity of a function at a point
paragraph_text: Continuity of a function at a point
  
  The definition of continuity at a point is given through limits.
  
  The above definition of a limit is true even if f (c) ≠ L {\displaystyle f(c)\neq L}. Indeed, the function f need not even be defined at c. However, if f (c) {\displaystyle f(c)} is defined and is equal to L {\displaystyle L}, then the function is said to be continuous at the point c {\displaystyle c}.
  
  Equivalently, the function is continuous at c {\displaystyle c} if f (x) → f (c) {\displaystyle f(x)\rightarrow f(c)} as x → c {\displaystyle x\rightarrow c}, or in terms of sequences, whenever x n → c {\displaystyle x_{n}\rightarrow c}, then f (x n) → f (c) {\displaystyle f(x_{n})\rightarrow f(c)}.
  
  An example of a limit where f {\displaystyle f} is not defined at c {\displaystyle c} is given below.
  
  Consider the function
  
  f (x) = x 2 − 1 x − 1. {\displaystyle f(x)={\frac {x^{2}-1}{x-1}}.}
  
  then f (1) is not defined (see Indeterminate form), yet as x moves arbitrarily close to 1, f (x) correspondingly approaches 2:
  
  Thus, f (x) can be made arbitrarily close to the limit of 2—just by making x sufficiently close to 1.
  
  In other words, lim x → 1 x 2 − 1 x − 1 = 2. {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=2.}
  
  This can also be calculated algebraically, as x 2 − 1 x − 1 = (x + 1) (x − 1) x − 1 = x + 1 {\textstyle {\frac {x^{2}-1}{x-1}}={\frac {(x+1)(x-1)}{x-1}}=x+1} for all real numbers x ≠ 1.
  
  Now, since x + 1 is continuous in x at 1, we can now plug in 1 for x, leading to the equation lim x → 1 x 2 − 1 x − 1 = 1 + 1 = 2. {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=1+1=2.}
  
  In addition to limits at finite values, functions can also have limits at infinity. For example, consider the function f (x) = 2 x − 1 x {\displaystyle f(x)={\frac {2x-1}{x}}} where:
  
  As x becomes extremely large, the value of f (x) approaches 2, and the value of f (x) can be made as close to 2 as one could wish—by making x sufficiently large. So in this case, the limit of f (x) as x approaches infinity is 2, or in mathematical notation, lim x → ∞ 2 x − 1 x = 2. {\displaystyle \lim _{x\to \infty }{\frac {2x-1}{x}}=2.}