Continuity and Differentiability play an important role in calculus. In fact, they provide the foundation for the theory of calculus. Now, we will discuss these concepts in detail.

Continuity of a Function

We know that a function is said to be continuous at a point if it has no break at that point. This is possible if the values of the function in the neighbourhood of that point are extremely close to the limiting value of the function at that point. Hence, continuity of the function at a point is closely related to the limiting behaviour of the function at the point under consideration. We give below the definition of continuity in precise form.

Right and Left continuity of a function

Definition:  A real valued function f(x) is said to be left-continuous or continuous from the left if given >0, however small, there exists a + ve real number δ(∈) such that,

|f(x)-f(a)|< for a- δ <x<a

In symbols, we can say, f is left continuous at ‘a’

if f(x)  = f(a) 

similarly  a real valued function f(x) is said to be right-continuous or continuous from right at a iff given >0, however small, there exists a +ve real number δ(∈) such that

fx-fa< for a<x<a+ δ

In symbols, f is right continuous at ‘a’ if

f(x)  = f(a)

Continuity of a function at a point

A function f is said to be continuous at x = a if given > 0, however small, there exists a + ve real number δ () such that

|f(x)-f(a)|< for x – a<

In symbols, f is said to be continuous at a

iff f(x)  = f(a) 

Now, f(x)   exists

if f(x)  = f(x)   (both existing finitely)

Hence f(x) is continuous at a if it is left continuous as well as right continuous at a.

A function f which is not continuous at a point is said to be discontinuous at that point. 

Continuity of a function in an interval

(1) Open-Interval: A real valued function f defined on the open interval (a, b) is said to be continuous on (a, b) if f is continuous at x=c for all c (a, b).

(2) Closed –Interval: A real valued function f defined on the closed interval [a, b] said to be  continuous on [a, b] if

 (i) f is right continuous at a i.e.,  f(x)  = f(a) 

 (ii) f is left continuous at b i.e.,  f(x)  = f(a) 

 (iii) f is continuous at c c (a, b)

i.e. f(x) =f(c) c(a, b)

Remark: Such a function has continuous graphs on (a, b).

Continuous function: if a function f is continuous at every point of its domain, it is said to be a continuous function. 

Domain of continuity: The set of all points where the function is continuous is called the domain of continuity.

Algebra of continuous functions

If f, g are two continuous functions at a, then

(i) kf is continuous at a, k∈R

(ii) (f± g) is continuous at a

(iii) fg if continuous at a

(iv) fg is continuous at a, provided g (a) ≠ 0.

Differentiability

Definition: The derivative of a function f at a point x = a is defined by

f'(a) =  f(a+h)-f(a) h

provided the limit exists.

GEOMETRICAL SIGNIFICANCE OF DERIVATIVE

Let y=f(x) be a function defined in some -neighbourhood of c, where is a small positive real number. Let P (c, f(c)) and Q ((c+h), f(c + h)) be two neighbouring points on the graph of y=f(x), where c+h lies in -neighbourhood of c and his very small,h0.

Then slope of chord PQ

  f(c+h)-f(c) (c+h)-c = f(c+h)-f(c) h

Let Q approach P along the curve. The chord PQ becomes tangent at P in the limiting position when Q coincides with P.

Thus slope of the chord PQ becomes slope of the tangent at P when Q→P i.e., when h→0.

Thus the slope of the tangent at P = f(c+h)-f(c) h

This limit (if it exists) is called the derivative of ‘f‘ at c and is denoted by f'(c) or dydxx=c

RELATION BETWEEN DERIVABILITY AND CONTINUITY

If a real valued function ‘f‘ is finitely derivable at any point of its domain, it is necessarily continuous.

Proof:  Let the function ‘f’ be derivable at c.

Then f(x) is defined in some -neighbourhood of c and f(c+h)-f(c) h exists finitely.

Where 0<h<δ

 i.e.    f(c+h)-f(c) h = f'(c)

Clearly f'(c) also exists.

Now fc+h-fc=.h

∴  fc+h-fc = f(c+h)-f(c) h .h

⟹ f(c+h)-f(c) h .h  = f‘c.0=0

⟹fc+h=fc

fx =fc

Hence, f is continuous at c.

Conclusion

In this article, we have learned about the Continuity of function at a point, continuity over an open and closed intervals and relationship between continuity and Differentiability etc. Continuity and Differentiability plays an important role in calculus. In fact it provides the foundation for the theory of calculus. Knowing all these, the students will get a reasonably good idea about methods to check continuity and Differentiability of function at a point.