JAMIA 2025 — Mathematics PYQ

For a function to be continuous at $x=a$, the following three criteria must be met:

1. The function is defined at $a$:

   $f(a)$ must exist (it cannot be undefined or infinite).

2. The limit of the function exists as $x$ approaches $a$:

   $$\underset{x\to a}{\lim }f(x)\text{ exists}$$

   This implies that the Left-Hand Limit (LHL) must equal the Right-Hand Limit (RHL):

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

3. The limit value equals the function value:

   $$\underset{x\to a}{\lim }f(x)=f(a)$$

Mathematical Representation

In a single combined equation, a function $f(x)$ is continuous at $x=a$ if:

Summary Table

If any of these conditions fail, the function is said to be discontinuous at $x=a$.

Final Answer:

A function is continuous at $x=a$ if ${\lim }_{x\to a}f(x)=f(a)$.