Notes On Continuity of a Function - CBSE Class 12 Maths
Suppose f is a real function on a subset of real numbers. Let c be a point in the domain of f. Then f is continuous at c, if $\underset{x\to \text{c}}{\text{lim}}\text{f(x)}=\text{f(c)}$ . If f is not continuous at x = c, then we say that f is discontinuous at c, and c is called a point of discontinuity of function f. Example: The continuity of  f(x) = 3x + 1 at x = 1. =   = 3(1)+1 = 4 = 4 = f(1) Hence, f(x) = 3x + 1 is continuous at x = 1 Example: Value of  f(x) = 2 at x = -0.01, -0.001, -0.001.... ∴  $\underset{x\to {\text{0}}^{\text{-}}}{\text{lim}}\text{f(x)}=\text{2}$$\underset{}{\text{}}$ Value of f(x) = 3 at x = -0.01, -0.001, -0.001.... ∴  $\underset{x\to {\text{0}}^{\text{+}}}{\text{lim}}\text{f(x)}=\text{3}$$\underset{}{\text{}}$ f(0) = 2 f(0) =  ≠ $\underset{x\to {\text{0}}^{\text{+}}}{\text{lim}}\text{f(x)}$$\underset{}{\text{}}$ f(x) is not continuous at x = 0.

#### Summary

Suppose f is a real function on a subset of real numbers. Let c be a point in the domain of f. Then f is continuous at c, if $\underset{x\to \text{c}}{\text{lim}}\text{f(x)}=\text{f(c)}$ . If f is not continuous at x = c, then we say that f is discontinuous at c, and c is called a point of discontinuity of function f. Example: The continuity of  f(x) = 3x + 1 at x = 1. =   = 3(1)+1 = 4 = 4 = f(1) Hence, f(x) = 3x + 1 is continuous at x = 1 Example: Value of  f(x) = 2 at x = -0.01, -0.001, -0.001.... ∴  $\underset{x\to {\text{0}}^{\text{-}}}{\text{lim}}\text{f(x)}=\text{2}$$\underset{}{\text{}}$ Value of f(x) = 3 at x = -0.01, -0.001, -0.001.... ∴  $\underset{x\to {\text{0}}^{\text{+}}}{\text{lim}}\text{f(x)}=\text{3}$$\underset{}{\text{}}$ f(0) = 2 f(0) =  ≠ $\underset{x\to {\text{0}}^{\text{+}}}{\text{lim}}\text{f(x)}$$\underset{}{\text{}}$ f(x) is not continuous at x = 0.

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