Problems from Continuity of a function at a point

Let's look at the continuity of the function

$f (x) = {\begin{array}{rcl} x - 3 & if & x \neq 3 \\ 0 & if & x = 3 \end{array}$

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Development:

The functions that define $f (x)$ are continuous since they are polynomial, therefore they can only be discontinuous if the two functions do not connect when $x = 3$ .

$\begin{array}{l} lim_{x \to 3^{-}} f (x) = lim_{x \to 3} (x - 3) = 0 \\ lim_{x \to 3^{+}} f (x) = lim_{x \to 3} (x - 3) = 0 \\ f (3) = 0 \end{array}$ As the lateral limits coincide with the value of the function, the function is continuous.

Solution:

The function is continuous when $x = 3$ and in all $R$ .

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Let's look at the continuity of the function

$f (x) = {\begin{array}{rcl} x^{2} + 2 & if & x < 1 \\ 3 x & if & x \geq 1 \end{array}$

See development and solution

Development:

The functions that define $f (x)$ are continuous since they are polynomial, therefore they can only be discontinuous if they do not connect when $x = 1$ .

$\begin{array}{l} lim_{x \to 1^{-}} f (x) = lim_{x \to 1} (x^{2} + 2) = 1^{2} + 2 = 3 \\ lim_{x \to 1^{+}} f (x) = lim_{x \to 1} (3 x) = 3 \\ f (1) = 3 \cdot 1 = 3 \end{array}$ As the lateral limits coincide with the value of the function, the function is continuous.

Solution:

The function is continuous when $x = 1$ and in all $R$ .

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