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  lim x c f(x) = 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.

lim x 1 f(x)   = lim x 1 (3x+1)  = 3(1)+1 = 4

lim x 1 f(x)   = 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....

∴  lim x 0 - f(x) = 2

Value of f(x) = 3 at x = -0.01, -0.001, -0.001....

∴  lim x 0 + f(x) = 3

f(0) = 2

f(0) =  lim x 0 - f(x)    lim x 0 + f(x)

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  lim x c f(x) = 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.

lim x 1 f(x)   = lim x 1 (3x+1)  = 3(1)+1 = 4

lim x 1 f(x)   = 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....

∴  lim x 0 - f(x) = 2

Value of f(x) = 3 at x = -0.01, -0.001, -0.001....

∴  lim x 0 + f(x) = 3

f(0) = 2

f(0) =  lim x 0 - f(x)    lim x 0 + f(x)

f(x) is not continuous at x = 0.

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