`\color{green} ✍️` If the left hand limit, right hand limit and the value of the function at `x = c` exist and equal to each other, then f is said to be continuous at `x = c`.
`color{orange}{lim_(x->c^-) f(x) =f(c) = lim_(x->c^+) f(x)}`
`"● Note :"` if the right hand and left hand limits at `x = c` coincide, its said the common value is the limit of the function at `x = c`.
● If `f` is not continuous at `c`, we say `f` is discontinuous at `c` and `c` is called a point of discontinuity of `f`.
E.g., Let's examine the function `f` given by `f (x) = x^2` is continuous at `x = 0` or not?
● First note that the function is defined at the given point `x = 0` and its value is `0`.
Then we find the limit of the function at `x = 0`. Clearly
For `c>0` `=>` `lim_(x-> 0^+) f(x) = lim_(x-> 0) x^2=0^2=0`
For `c<0` `=>` `lim_(x-> 0^-) f(x) = lim_(x-> 0) x^2=0^2=0`
● Thus `lim_(x-> 0) f(x) =0 =f(x)`
Hence, `f` is continuous at `x = 0`.
Now let's another example
(ii) E.g., the continuity of the function `f` given by `f(x) = | x |` at `x = 0`.
Here, By definition : `f(x) ={ tt(( -x, if, x < 0),(x, if , x ge 0))`
● Clearly the function is defined at `0` and `f (0) = 0`. Left hand limit of `f` at `0` is
`=> lim_(x-> 0^(-)) f(x) =lim_(x-> 0^(-)) (-x) =0`
Similarly, the right hand limit of `f` at `0` is
`=> lim_(x->0^+) f(x) =lim_(x-> 0^+) x=0`
Thus, the left hand limit, right hand limit and the value of the function coincide at `x = 0`. Hence, `f` is continuous at `x = 0`.
`\color{green} ✍️` If the left hand limit, right hand limit and the value of the function at `x = c` exist and equal to each other, then f is said to be continuous at `x = c`.
`color{orange}{lim_(x->c^-) f(x) =f(c) = lim_(x->c^+) f(x)}`
`"● Note :"` if the right hand and left hand limits at `x = c` coincide, its said the common value is the limit of the function at `x = c`.
● If `f` is not continuous at `c`, we say `f` is discontinuous at `c` and `c` is called a point of discontinuity of `f`.
E.g., Let's examine the function `f` given by `f (x) = x^2` is continuous at `x = 0` or not?
● First note that the function is defined at the given point `x = 0` and its value is `0`.
Then we find the limit of the function at `x = 0`. Clearly
For `c>0` `=>` `lim_(x-> 0^+) f(x) = lim_(x-> 0) x^2=0^2=0`
For `c<0` `=>` `lim_(x-> 0^-) f(x) = lim_(x-> 0) x^2=0^2=0`
● Thus `lim_(x-> 0) f(x) =0 =f(x)`
Hence, `f` is continuous at `x = 0`.
Now let's another example
(ii) E.g., the continuity of the function `f` given by `f(x) = | x |` at `x = 0`.
Here, By definition : `f(x) ={ tt(( -x, if, x < 0),(x, if , x ge 0))`
● Clearly the function is defined at `0` and `f (0) = 0`. Left hand limit of `f` at `0` is
`=> lim_(x-> 0^(-)) f(x) =lim_(x-> 0^(-)) (-x) =0`
Similarly, the right hand limit of `f` at `0` is
`=> lim_(x->0^+) f(x) =lim_(x-> 0^+) x=0`
Thus, the left hand limit, right hand limit and the value of the function coincide at `x = 0`. Hence, `f` is continuous at `x = 0`.