Continuity of Functions and Types of Discontinuity | NIMCET

Quick answer: A function f is continuous at a if f(a) is defined, $\lim_{x \to a}f(x)$ exists, and the two are equal, that is, LHL = RHL = f(a). If the limit exists but differs from f(a) (or f(a) is undefined), the discontinuity is removable; if LHL ≠ RHL, it is a jump; if a one-sided limit is infinite, it is an infinite discontinuity. The intermediate value theorem says a continuous function on [a, b] takes every value between f(a) and f(b).

This builds on one-sided limits.

Continuity at a point

f is continuous at $x = a$ when

$\lim_{x \to a^-}f(x) = \lim_{x \to a^+}f(x) = f(a)$

f is continuous on an interval if it is continuous at every point of it (with one-sided continuity at closed endpoints).

Types of discontinuity

A removable discontinuity can be fixed by redefining f(a) as the limit.

Continuity of standard functions

Combining continuous functions

If f and g are continuous at a:

A sum of a continuous and a discontinuous function is discontinuous; a sum of two discontinuous functions may be continuous.

Intermediate value theorem

If f is continuous on [a, b] and k lies between f(a) and f(b), then $f(c) = k$ for some c in [a, b].

Consequence: if f(a) and f(b) have opposite signs, f has a root in (a, b).

Worked examples

Example 1: finding a constant

Find k if $f(x) = \frac{\sin 3x}{x}$ for $x \ne 0$ and $f(0) = k$ is continuous at 0.

$\lim_{x \to 0}\frac{\sin 3x}{x} = 3$, so $k = 3$.

Example 2: piecewise function

$f(x) = ax + 1$ for $x \le 2$ and $f(x) = x^2 - 1$ for $x > 2$. Find a for continuity.

LHL = f(2) $= 2a + 1$, RHL $= 3$, so $a = 1$.

Example 3: type of discontinuity

Classify the discontinuity of $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$.

The limit is 4 but f(2) is undefined, so the discontinuity is removable.

Example 4: points of discontinuity

Where is $f(x) = [x]$ discontinuous on $(0, 3)$? At $x = 1$ and $x = 2$.

Example 5: IVT

Show that $x^3 - x - 1 = 0$ has a root in (1, 2).

$f(1) = -1 < 0$ and $f(2) = 5 > 0$, and f is continuous, so a root lies between 1 and 2.

Summary table

Exam tips

• For piecewise functions, check only the joining points; each piece is usually continuous.
• To find a constant for continuity, equate the limit to f(a).
• Combine with standard limits from L'Hospital's rule and standard limits.

Next, study differentiability and left and right hand derivatives, or see the NIMCET Calculus chapter.

Common mistakes

• Checking only that the limit exists, without comparing it with f(a).
• Assuming a function undefined at a point can be continuous there.
• Treating $\lvert x\rvert$ as discontinuous at 0 (it is continuous but not differentiable).
• Forgetting that $\tan x$ is discontinuous at odd multiples of $\frac{\pi}{2}$.

Frequently Asked Questions

When is a function continuous at a point?

When f(a) is defined and the left-hand limit, right-hand limit and f(a) are all equal.

What is a removable discontinuity?

A point where the limit exists but is not equal to f(a), or f(a) is undefined. Redefining f(a) removes it.

Is the greatest integer function continuous?

It is continuous everywhere except at integers, where it has jump discontinuities.

What does the intermediate value theorem say?

A continuous function on [a, b] takes every value between f(a) and f(b).

Is |x| continuous at 0?

Yes. It is continuous at 0, though it is not differentiable there.