Q. 84 - Calculus

Question

Use what you know about one-sided limits to prove that a function $f$ is continuous at a point $x=c$ if and only if it is both left and right continuous at $x=c$.

Step-by-Step Solution

Answer

Ans:  If LHL (Left Hand Limit) =RHL(Right Hand Limit) At point $x=c$ then, function $f$ is continuous at a point $x=c$.

Step 1. Given information.

given, both left and right continuous at $x=c$

Step 2. Find LHL and RHL at point x = c .

Finding limits at point $x=c$

LHL $=\underset{x\to c^{-}}{lim} f\left(x\right) =c$

RHL $=\underset{x\to c^{+}}{lim} f\left(x\right)=c$

Step 3. Proof.

Since, LHL =RHL

Then, we can say that $f$ is continuous at a point $x=c$.