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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.
A real-valued function f(x)is said to be continuous at a point x=x_\circ in the domain if –
\lim_{x\to x_\circ} f(x) exists and is equal to f(x_\circ).
If a function f(x) is continuous at x=x_\circ then-
\lim_{x\to x_\circ ^+} f(x) = \lim_{x\to x_\circ ^-} f(x) = \lim_{x\to x_\circ} f(x)
Functions which are not continuous in nature are known as discontinuous.

Example 1 – For what value of \lambda is the function defined by

   \[ f(x)=\left \{ \begin{tabular}{ll} \lambda(x^2-2),&\:if\:x\leq0 \\ 4x+1,& otherwise\\ \end{tabular} \]

continuous at x=0

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 x=0

f(0) = \lambda * (0-2) = -2\lambda

Right-hand limit –

=\:\lim_{x\to 0^+} 4x+1=\:1

RHL equals the value of the function at 0 –

-2\lambda = 1

\lambda = \frac{-1}{2}

Example 2 – Calculate all the points of discontinuity of the function fthat are defined by –


Solution – The probable points of discontinuity can be written as x=0,1 since the sign of the modulus would change at these points.

For continuity at x=0,


=\lim_{x\to 0^-} |x|-|x-1|=\lim_{x\to 0^-} -x-(-(x-1))=\lim_{x\to 0^-} -x+x-1= -1


=\lim_{x\to 0^+} |x|-|x-1|=\lim_{x\to 0^+} x-(-(x-1))=\lim_{x\to 0^+} x+x-1= -1

Value of f(x) at x=0,

f(0) = 0-|0-1| = -1

Since LHL = RHL = f(0), the function is continuous at x=0For continuity at x=1,


=\lim_{x\to 1^-} |x|-|x-1|=\lim_{x\to 1^-} x-(-(x-1))=\lim_{x\to 1^-} x+x-1= 1


=\lim_{x\to 1^+} |x|-|x-1|=\lim_{x\to 1^+} x-(x-1)=\lim_{x\to 1^+} x-x+1= 1

Value of f(x) at x=1,

f(0) = 1-|1-1| = 1

And as LHL = RHL = f(1), the function would be continuous at x=1

Therefore, there is no point of discontinuity.

Understanding Differentiability 

The derivative of a real-valued function f(x) wrt x is the function f^\prime(x) and can be described as –
\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}

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 x=c,
\lim_{h\to 0} \frac{f(c+h)-f(c)}{h} 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, f(x) = |x| is continuous at x=0 but it is still not differentiable at that point.

Continuity at a Point

When we say that the function f is continuous at the point k, it needs to satisfy the following three conditions:
(i) Function is defined at given point, i.e.f(k) exists.

 The limit of f as x tend k exists, i.e.
lixk f(x) is present.

 This limit and the value of the function at k should be equal, i.e.
limxk 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.

lixk [f(x)f(k)

lixk [(xk) * f(x)f(k) / (xk)

lixk (xk) lix[f(x)f(k) / (xk)]

0. f(k)

Therefore, we have

lixk [f(x)f(k)]= 0

lixk f(x)=limx→k f(k)

This was our article on continuity and differentiability. For more such articles, keep following us here.

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