Meaning of continuity of function –
If we talk about a general meaning of continuity of a function f(x) , we can say that if we plot the coordinates (x , f(x)) and try to join all those points in the specified region, we can do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve without any breakage.
Mathematically we define the same thing as given below :
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to be checked
If :–
\(\lim\limits_{h \to 0} f(c-h)\) = \(\lim\limits_{h \to 0} f(c+h)\) = f(c) .... equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means :–
Limiting value of the left neighbourhood of x = c also called left hand limit LHL {i.e, \(\lim\limits_{h \to 0} f(c-h)\)} must be equal to limiting value of right neighbourhood of x = c called right hand limit RHL {i.e, \(\lim\limits_{h \to 0} f(c+h)\)} and both must be equal to the value of f(x) at x = c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds true, function is continuous else it is discontinuous.
Given,
\(f(x) = \begin{cases} \frac{x^2-3x+2}{x-1}&, \quad if\, x ≠1\\k &, \quad if\, x=1 \end{cases} \)….equation 2
We need to find the value of k such that f(x) is continuous at x = 1
Since f(x) is continuous at x = 1
∴ (LHL as x tends to 1) = (RHL as x tends to 1) = f(1)
As,
f(1) = k [from equation 2]
We can find either LHL or RHL to equate with f(1)
Let’s find RHL,you can find LHL also.
As,
f(x) is continuous
∴ RHL = f(1)
∴ k = –1
