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# Question:Is the function $f$ defined by:$f(x) = \begin{cases} x & \text{if } x \leq 1 \\ 5 & \text{if } x > 1 \end{cases}$​ continuous at $x = 0\ ?\ At\ \ x = 1\ ?\ At\ \ x = 2$? Find all points of discontinuity of $f$.

Last updated date: 14th Aug 2024
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Hint:

A function is continuous at a point $x = c$ if:

1. $f(c)$ is defined.

2. $lim_{{x \to c^-}} f(x)$ (left-hand limit) exists.

3. $lim_{{x \to c^+}} f(x)$ (right-hand limit) exists.

4. $lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x) = f(c)$.

Step-by-step Solution:

1. At $x = 0$:

• $f(0)$ is clearly defined as 0.

• Left-hand limit: $lim_{{x \to 0^-}} f(x) = 0$

• Right-hand limit: $lim_{{x \to 0^+}} f(x) = 0$

• Since both the limits are equal to $f(0)$, the function is continuous at $x = 0$.

2. At $x = 1$:

• $f(1)$ is defined as 1.

• Left-hand limit: $lim_{{x \to 1^-}} f(x) = 1$

• Right-hand limit: $lim_{{x \to 1^+}} f(x) = 5$

• The left-hand limit is not equal to the right-hand limit, so the function is discontinuous at $x = 1$.

3. At $x = 2$:

• $f(2)$ is defined as 5.

• Left-hand limit: $lim_{{x \to 2^-}} f(x) = 5$

• Right-hand limit: $lim_{{x \to 2^+}} f(x) = 5$

• Both the limits are equal to $f(2)$, so the function is continuous at $x = 2$.

Points of Discontinuity:

A function is discontinuous at a point if it is not continuous at that point. Therefore, the only point of discontinuity of $f$ is $x = 1$.

Note:

The function $f(x)$ is continuous everywhere except when it changes its definition rule, which is $x = 1$. Whenever you encounter piecewise-defined functions, always check the points where the function definition changes to find possible points of discontinuity.