Proving Function Continuity Using Limits: A Step-By-Step Guide

f is continuous at x=a

f(a) exists, lim x->a f(x) exists, lim x->a f(x) = f(a)

To show that a function f is continuous at x=a, we need to demonstrate that as x approaches a, the limit of f(x) exists and is equal to the value of f(a).

In mathematical terms, we want to prove that:

`lim_(x->a) f(x) = f(a)`

Here are some steps we can follow to demonstrate continuity:

1. Check that f(a) is defined (i.e. the function is defined at x=a).

2. Evaluate `lim_(x->a) f(x)`. This involves taking the limit of f(x) as x approaches a from both the left and the right sides of a. If these two limits are equal to each other and equal to f(a), then the limit exists and is equal to f(a).

3. Since we have shown that the limit of f(x) exists and is equal to f(a), we can conclude that f is continuous at x=a.

Note that to show continuity at a, we need to check both the limit from the left and the limit from the right, since the limit itself may be defined but not equal to f(a) if the two one-sided limits are not equal.

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