Continuity and Differentiability:
Continuity and Differentiability is an important topic in class 11th and 12th calculus, usually fetching a lot of questions in JEE Main and Advanced. In this article, read about continuity and differentiability for commerce students.
Understanding Continuity
If the graph of a function is a single unbroken curve, then it is called continuous over a range.
Ideally,
A real-valued function 
is said to be continuous at a point 
in the domain if –
exists and is equal to 
.
If a function is continuous at then-
Functions which are not continuous in nature are known as discontinuous.
Example 1 – For what value of 
is the function defined by
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continuous at 
Solution – A function can be continuous only when the left-hand limit, right-hand limit and the value of the function at that point are equal.
Value of function at
Right-hand limit –
RHL equals the value of the function at 0 –
Example 2 – Calculate all the points of discontinuity of the function 
that are defined by –
.
Solution – The probable points of discontinuity can be written as 
since the sign of the modulus would change at these points.
For continuity at ,
LHL-
RHL
Value of at ,
Since LHL = RHL = 
, the function is continuous at For continuity at 
,
LHL-
RHL
Value of at ,
And as LHL = RHL = 
, the function would be continuous at
Therefore, there is no point of discontinuity.
Understanding Differentiability
The derivative of a real-valued function wrt 
is the function 
and can be described as –
A function is called differentiable in a situation where the derivative of this particular function exists at all points of its domain. In order to check the differentiability of a function at the point 
,
needs to exist.
However, a function is also continuous at a point when it is differentiable at that point.
Note – When a function is continuous at a point, it does not mean that is also differentiable at that point. For instance, 
is continuous at but it is still not differentiable at that point.
Continuity at a Point
(ii) The limit of f as x tend k exists, i.e.lim x→k f(x) is present.
(iii) This limit and the value of the function at k should be equal, i.e.limx→k f(x) = f(k).The function f would be called discontinuous or not continuous at point k when any of the above three conditions are not valid.
Continuity in an Interval
We can say that it is a continuous function of an interval when it is continuous at each and every point of the interval. It simply means that all three conditions of continuity need to be fulfilled at all the points within the given interval.
If (a, b) be an open interval, then a function f(x) would be called continuous in the interval (a, b) if f is proved to be continuous at its each and every point.
Always remember that when we conclude that a function is continuous, it signifies that the function is continuous at all the real numbers. In this case, we would also be able to say that the function is continuous on the set of real numbers or over the interval (−∞,∞−∞,∞). But we just say “continuous” for our convenience.
Differentiability Implies Continuity
For proving the fact that differentiability implies continuity, we would be proving the second relation with the help of the first relation.
lim x→k [f(x)−f(k)]
lim x→k [(x−k) * f(x)−f(k) / (x−k)
= lim x→k (x−k) lim x→k [f(x)−f(k) / (x−k)]
= 0. f′(k)
Therefore, we have
lim x→k [f(x)−f(k)]= 0
lim x→k f(x)=limx→k f(k)
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