Continuity And Differentiability For Class 12

Definition of Continuity:

(i) The continuity of a real function (f) on a subset of the real numbers is defined when the function exist at point c and is given as-

\(\lim\limits_{x \to c}f(x) = f(c)\)

(ii) A real function (f) is said to be continuous if it is continuous at every point in the domain of f.

Consider a function f(x), the function is said to be continuous at every point in [a, b] including the end points a and b.

Continuity of f at a means,

\(\lim\limits_{x \to a}f(x) = f(a)\)

Continuity of f at a means,

\(\lim\limits_{x \to b}f(x) = f(b)\)

Continuity And Differentiability For Class 12

Algebra of continuous functions:

If the two real functions, say f and g, are continuous at a real number c, then

(i) f + g is continuous at x=c.

(ii) f – g is continuous at x=c.

(iii) f. g is continuous at x=c.

(iv) \(\frac{f}{g}\) is continuous at x=c.’

Differentiability formula-

The derivative of a function f at c is defined by-

\(\lim\limits_{h \to 0} \frac{f(x+h) – f(c)}{h}\)


Practise This Question

If x=2costcot2t,y=2sintsin2t, then d2ydx2 at t = π2 is